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On allocations attainable through Nash equilibria
JOURNAL
OF ECONOMIC
THEORY
On Allocations
21, 140-165 (1979)
Attainable
through
Nash Equilibria*
LEONID HURWICZ+ Department of Economics, University of Minnesota, Minneapolis, Minnesota 55455 Received July 28, 1977; revised December 14, 1978
The purpose of this paper is to show that there are severe limitations on allocations attainable through mechanisms which can be viewed as Nash noncooperative games in which the agents’ messages constitute their strategies. In such mechanisms, the designer chooses the individual agents’ message sets and the rule (called outcomefunctionl) specifying the resource allocation resulting from the agents’ message choices. The messages chosen by agents constitute a Nash (noncooperative) equilibrium if no agent can unilaterally improve his situation as long as others do not change their messages. Clearly the equilibrium messages will depend on the agents’ characteristics (initial endowments, preferences, technology) and, hence, so will the allocations corresponding to these messages. We shall refer to the totality of all the agents’ characteristics as the environment and we shall call the allocations generated by Nash equilibrium messages Nash allocations. The relationship between environments and the corresponding Nash allocation sets specified by an outcome function constitutes the (Nash) performance correspondenceaassociated with that outcome function. The designer of the mechanism is assumed to be interested in the performance correspondence, but he can only control it indirectly by choosing the outcome function and the message sets. This paper shows that such indirect control is quite limited: only certain types of performance correspondences can be generated by a Nash equilibrium mechanism even if the designer has great freedom in selecting outcome functions and message sets. The economist’s interest in the design of resource allocating mechanisms * In addition to a general intellectual indebtedness in this area of analysis to T. Groves, J. Ledyard, S. Reiter, and D. Schmeidler, I also want to acknowledge helpful conversations with .I. Jordan (who pointed out an error in an earlier version of this paper), A. Postlewaite, R. Radner, L. Shapiro, S. Smale, and W. Thomson, The appointment as visiting Ford professor in the Economics Department at Berkeley (U. of Calif.) provided an ideal setting for this research, with much of the stimulus due to M. Kun and the summer IMSSS seminars at Stanford. + Project supported by NSF Grants GS-31276X’and SOC 76-14786. 1 Also called “game form” by other writers. a We follow here the formulation due to Mount and Reiter [7]. “Social choice correspondence” is another term used.
140 0022-0531/79/040140-26$02.00/O Copyright All rights
0 1979 by Academic Press, Inc. of reproduction in any form reserved.
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was greatly stimulated by the “free rider” problem arising in connection with public goods. In line with the prevailing tradition, interest in this area was focused on Pareto-optimality. Groves and Ledyard [3], whose model played a crucial role in stimulating the present line of research, constructed outcome functions whose Nash allocations were always optimal, but not necessarily individually rational; i.e., there is no guarantee that the equilibrium satisfaction levels of agents are at least as high as those at initial endowments. It became natural to inquire whether a different choice of outcome functions would yield a performance both Pareto-optimal and individually rational for all economies of a specified class. One way to accomplish this was to find outcome functions yielding the Lindahl performance, since the latter is both Pareto-optimal and individually rational. Such outcome functions, both balanced and not balanced, were shown to exist in Hurwicz [5] for economies involving both public and private goods, where the latter can be either consumed or used as inputs to produce public goods under constant returns. However, the Lindahl correspondence is only one among many that guarantee Pareto-optimal and individually rational allocations. It is therefore natural to ask whether other correspondences with the two attributes of Pareto-optimality and individual rationality can also be realized through Nash equilibria. Section 2 of the present note3 provides a partial answer to this question. Theorem 3 shows that, under certain assumptions on the environment class covered, and assuming a continuity property for the performance correspondence, if all Nash allocations for a given environment are Pareto-optimal and individually rational, then every Lindahl allocation for that environment is among its Nash allocations. Theorem 4 constitutes a partial converse. It states that, again under certain assumptions on the environment class covered, and assuming a convexity property for the outcome functions, every interior Nash allocation is a Lindahl allocation for the given environment. The nature of the convexity property of the outcome functions is exemplified in the Corollary4 to Theorem 4 which implies that the following is suficient: for each participant, the outcome function is concave in this individual’s own message (strategy). However, the convexity condition of Theorem 4 is significantly weaker. Of course, in Theorem 4 and its Corollary it is again postulated that every Nash allocation is Pareto-optimal and individually rational for the given environment. 8 Although historically the issues of this paper arose primarily in connection with public goods, we find it convenient in the body of this paper to reverse the order and to start with private goods (“Edgeworth Box”) economies. ‘The Corollary to Theorem 4 is an exact analogue of the Corollary to Theorem 2, with an identical proof. Only the Corollary to Theorem 2 and its proof are presented formally.
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LEONID
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Assumptions on the environment are of two types. On the one hand, they require that every economy belonging to the class covered have certain properties. On the other hand, they require that the class covered be sufficiently rich; in particular, that it contain certain special linear and strictly concave economies described at the end of the section on notation. As for requirements of the first type, both theorems assume nonnegative orthant as consumption set and strictly positive initial endowments, as well as preferences strictly increasing in every good. In addition, Theorem 4 and the Corollary postulate continuous convex preferences and refer to interior allocations only. It may be noted that the assumptions on the performance correspondence, including the continuity property (closed graph) mentioned above, are satisfied by the Lindahl correspondence. Hence our results are not vacuous. On the other hand, one can easily think of reasonable performance correspondences that are excluded by these results. One example is given by a performance correspondence whose allocations maximize the product of utility gains from the initial level (the Nash bargaining solution). Another, by a performance correspondence whose allocations maximize utility gains subject to the constraint that these gains be equal for all participants (or satisfy some other a priori imposed ratio requirements). In the latter two examples it is to be understood that some canonical utility indices have been specified.5 It appears that the Groves-Ledyard outcome functions satisfy the hypotheses of our two Theorems. Since they do not yield the Lindahl correspondence and do guarantee Pareto-optimality, it is not surprising that individual rationality is violated. Analogous results are obtained for economies in which public goods are absent. In Section 1, we consider pure exchange economies with private goods only, For such economies, outcome functions realizing the Walrasian performance correspondence through Nash Equilibria were first constructed by Schmeidler [9] and later Hurwicz [5, 61. Again, the question arises what other performance correspondences, guaranteeing Pareto-optimality and individual rationality, could be realized through Nash Equilibria. For cases so far investigated, the answers are analogous to those obtained for the public goods model. In particular, Theorem 1 shows that, given a continuity property of the performance correspondence, and certain assumptions on the class of environments covered, if all Nash allocations for a given environment are Pareto-optimal and individually rational, then every Walrasian allocation for that environment is a Nash allocation. Theorem 2 again constitutes a partial converse of Theorem 2. Under 5 For instance, in a classical environment let each agent’s utility of a given bundle z’ be measured by the distance from the origin 0 to the point z” which is on the same indifference surface as z’ and which contains the same quantities of all goods; i.e., z” -i z’, z = (z; ,..., z;,, z: = 1.. = z; , and u*(z’) = ((z;)~ + ... + (z;)~)~/~.
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assumptions analogous to those of Theorem 4, every interior Nash Equilibrium allocation is Walrasian. The Corollary to Theorem 2 shows that the convexity condition in Theorem 2 is satisfied when every individual’s outcome function is concave in this individual’s message (strategy). However, the convexity condition of Theorem 2 requires much less than the concavity of outcome functions. In Section 3 we extend the results of Section 1 to economies with production although without public goods. (A further extension to economies with production and public goods is possible.) Theorems 1” and 2* are the respective counterparts of Theorems 1 and 2. It should be noted that when production sets are convex but constants returns do not prevail everywhere, the appearance of “profits” necessitates ‘a reconsideration of the notion of individual rationality. This is so because it would not in general make sense to use the initial endowment as the reference point for utility comparisons. We deal with this problem by introducing the notions of a reference distribution and a guarantee structure, and speak of allocations that are individually rational with respect to specific guarantee structures. One such structure is that of fixed shares (defined in (1) of Section 3) which is analogous to a “private ownership economy” of the Arrow-Debreu type. Alternative guarantee structures would also deserve investigation, but this has not been done in the present note. It is rather remarkable that our results are valid whether or not the outcome functions are balanced away from equilibrium. Furthermore, they are valid not only for games proper but also for what we call quasi-games in which non-players may also participate. This allows for the introduction of auctioneers and “souless” production managers who do not have preferences but have decision rules. The extension of our results to quasi-games is discussed in Section 4. NOTATION
R R+ Iw++ W (resp. R+*, R:+)
The The The The
set of set of set of q-fold
all reals all nonnegative reals all positive reals Cartesian product of R (resp. of R+q, Rt+)
Environment n
N = (l,..., n} 1 6Ji Ri or &
642/2III-IO
The number of agents (n > 2) The set of agents The number of goods (I > 2) The i-th initial endowment; w = (wl,..., w-) (wi E R:+) The i-th preference relation (reflexive); R = (Rl,..., R”); (Ri C (w+I x [w+I)
144
LEONID
Pi or >$ -i ed = (cd, Ri)
HURWICZ
The i-th strict preference relation The i-th indifference relation The i-th characteristic; e = (el,..., e”) an environment (an economy) A class of pure exchange economies e with [w,’ as consumption sets
E Allocations zi
Net increment in commodity holdings by individual i (zi E W) An allocation The set of allocations Pareto-optimal for e The set of allocations “individually rational” for e [i.e., allocations z such that, for each i E N, wi + ziRiwi where e = ((wi, Ri))i6N]
2 = (z’,..., z”) 9(e) C Rzn X(e) C Rzn
S(e) = .9(e) n 9(e) W(e) E Rz+ln
The set of pairs (p, z) constituting Walrasian Equilibria for e (p * zi = 0 for each i E N) The set of allocations Walrasian for e; W(e) = {z: (p, z) E W(e) for some p}
We> The Game &vi JpiC
The i-th message (strategy) space =A1
x
h”: ud! -+ [wz
.. . x
&i-l
x
di+l
x
. .. x
4%
The i-th outcome function (its values are net increments in commodity holdings by i-th individual) we write zi = hi(mi , m)i() where mi E &Ii and m’i’ g &?pc The outcome function h = (hl,..., h”) hi(Mi, rnji() = {zi: zi = hi(mi , rnji’) for some mi E &!i} where rnji( is a fixed element of JZ)ic Hi(m) = {zi: zi < hi(mi , rnji() for some mi E d@>, rn)i’ E JY)Q vh(e) = {m*: Vi E N, oi + h”(m*) Riwi + hi(mi , m*ji() for all mi E Xi} where e = ((wi, Ri))ieN = the set of Nash Equilibrium message n-tuples for the economy e when the outcome function h is used v,,:E+.& The Nash Equilibrium message correspondence &(e) = {(zl,..., zn) E !Rzn: Vi E N, zi = hi(m) for some m E vh(e)} = the set of Nash Equilibrium allocations for e when h is used The Nash performance correspondence when h is A$: E -+ (Wzn used Jh = h 0 vh
ALLOCATIONS
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Certain Special Environments x IR,.z* R ‘is a preference relation generated by a utility function of the form v(x, y) = x + /? . y for some /3 E lRq;;‘> where x E R and y E Rz-l
@)L=(RCR+Z
ELi(oi) = {ei: ei = (cd, R”), Ri E 9?,.> EL(w) = {(el,..., en): Vi E N, ei E ELi(wi)} 9?c = {R C R+z x R+z: R is a preference relation
generated by a utility function of the form u(x, y) = w. + x + (1 - 0)~ * (oy + y) + with 0<8<1; q$,k,>O for s= 8 CL’, qs ln(w, +yg+kk,) 1,..., 1 - 1; & E [FB:::) E RZ-l where W, E R,, , XE!R,WyER~~,Y
Eci(wi) = (ei: es = (cod,Ri), Ri E a,-} E,(w) = {(el,..., en): Vi E N, ei EEci(w”)} h(w) = W,..., en): Vi E iV, ei E ELi(wi) U Eci(w”)}.
1. PURE EXCHANGE
ECONOMIES WITHOUT
PUBLIC
GOODS
1.1. THEOREM 1. Let E be a class of pure exchange economies whose elementse satisfy the following conditions: (a) n>,2; (b) l 3 2; (c) for each i E iV, (c-1) d E lR;+ , (c.2) the admissibleconsumptionset is Iw+I; (c.3) the preference relation R” is strictly increasing in each good;
(d) ifE contains B = ((&, e))i.N and Z E J?&(G) where (I, = (G,..., 3) then e’E E. Furthermore, let the outcomefunction h have the folIowing properties:
(i) for each e E E, .4(e) C S(e);
(ii)
zye E E,,-(W) C E, then 4(e)
(iii)
# 0;
ifELc(&) C B; e, E EL,-(&), z” E &(e,), IJ = 1, 2 ,...; e. E W-3;
then z, E J$(eo).
e, = lim e, ;
z, = lim z, ;
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LEONID HURWICZ
Then X$(e) 2 W(e)
for all
e E E.
ProojI Let 2 = ((69, ai)),, belong to E. We shall show that every Walrasian allocation for 2 is also a Nash allocation, i.e., that Z E W(Z) implies P E .4$(Z). Let then ( 3, 2) be a Walrasian equilibrium for 2. Since 62 > 0 by (cl), there is a good j such that ojl + Zjl > 0. We shall denote this j-th good by x and other goods by ys, s = I,..., I- 1, so that 6,’ + X1 > 0. We shall also write zi = (xi, yi) and p = ( & , &,) = (1, &,). Such normalization of prices is legitimate, since, by (c.3), p. > 0. Let r = I,..., IZ - 1, and consider a sequence of characteristics v = 1, 2,...,
eYT = (67, R,?),
where RUTis generated by the utility function6 U”‘(X’, y’) = WzT+ XT + (1 - e,) py . (G*r + y’) Z-1 +
0,
1 s=1
9;
ld4,
+
Y,7
+
k,')
and k,’ > 0 for all s = I,..., I-l;O~&,~1;lim8,=0;and 4.: = PY (6s + js’ + k,‘) for s = l,..., 1 - 1. Using the sequences eVr, r = I,..., n - 1, of characteristics just defined, we now construct a sequence e, = (e,l,..., e:-‘, eLn) of economies where e,” = (w”, RLn) is generated by the utility function v” = (x”, y”) = 8,” + xa + p, * (f&n + y”) = p * (3 + 2”). Observe that lim e,’ = eLr = (G”, R,?), where RLT is generated p * (6~ + z?). Therefore
r = l,..., n - 1,
by the same linear
utility
lim e, = e, = (eL1,..., eLn),
function
v’(z’) = (1)
an economy whose initial endowments are the same as in 2 and whose indifference sets are planes parallel to the Walrasian price (budget) plane through 5. 6 Other types of utility functions could also be used. Slightly different construction may be preferable when f is on the boundary.
ALLOCATIONS
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We shall now show that Z is a Nash allocation for the economy eL . In view of the “closed graph” property (iii), it will sufike to find a sequence of allocations z, such that zVis a Nash allocation for e, and lim z, = E
(2)
(See Fig. 1.)
FIG. 1. Arrows attached to indifference curves show the direction of preference. A symbol such as I? shows that the indifference curve so labeled corresponds to the preference relation R’ .
Since each economy ‘e, belongs to the class ELC , condition (ii) guarantees that it does have a Nash allocation. Furthermore, by (i), this allocation is located in the sets S(e,) of Pareto-optimal individually rational allocations. Now the sequence e, has been so constructed that the diameter of S(e,) -+ 0 as v ---f cc
(3)
and Z l S(e,)
for all
Therefore any sequence of allocations satisfy (2).
v = 1, 2,... .
(4)
zV such that z, is Nash for e, must
14%
LEONID
HURWICZ
To see that (3) and (4), hold note first that, for each v, an allocation w, Y%N is Pareto optimal for e, only if y’ = p” for all i E N. This is so because, for v = 1,2 ,... and s = l,..., I - 1,
and
auyay,r Iv+ = py, for r = 2,..., n. It follows that 5 is Pareto-optimal for every e, . As for individual rationality, it is seen that within the Pareto-optimal set @(e,) -where yi = jY for each i E N-we have ~,i(z,t) > u:(O) for all i EN if and only if
x,n a P and for all
5’ 2 5s.’
r = l,..., n - 1
where e is defined by the equality
or, explicitly,
In fact, the above inequalities on the xf together with the conditions yi = .j” for all i E N and the requirements of feasibility (staying in the Edgeworth Box) define the set S(e,) for each v. Since &;’ < Kr for all v, it follows that Z is both Pareto-optimal and individually rational in every economy of the sequence, so that (4) is satisfied. Noting that X* = -& * y’ for all i E N, we see from the formula for 5,’ that .&;’ + XT as v-00, r = I,..., n - 1 since the 0, term drops out asymptotically. Furthermore, since the x:, as well as the + add up to zero, it follows from the inequalities on the xV7that n-1 c #=l
m-1 x,7
<
c r-1
x7.
Using relations involving the 5,’ we conclude that x,’ + Z. This implies that x::; (xy’ - 57) + 0, and hence, x,” -+ Zn, Thus x, -+ Z, and hence z, = (xy , J) + (Z, J) = 1. So (2) has been established. (Of course, (3) is implicit in this argument.)
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149
EQUILIBRIA
So far we have shown that
This means that, for some m* E .A!, m* E vded
(5’)
Z = h(m*).
(5”)
and
Hence, because the indifference surfaces for each eL* are given by J * 3 = const., the fact that Z is a Nash allocation for eL implies that, for each i E N,
But then (6), in conjunction with (57, implies that m* E ~~(2). For, by hypothesis, (p, 2) is a Walrasian Equilibrium for .Z, and so every trade z” preferred to 3 (according to the preference relation I@) must cost more
\
ol
,5
-. %
FIG. 2. Shading such as d/ is placed on the inside of the boundary of the set /+(A’, m*“‘). Arrows attached to indifference curves show the direction of preference. A symbol such as I? shows that the indifference curve so labeled corresponds to the preference relation RI.
150
LEONID JXJRWICZ
than p - 9. (See Fig. 2.) Hence it cannot be reached by individual of the implication in (6). It follows that, for each i E N,
i in virtue
and, therefore,
This completes the proof of the Theorem. 1.2. THEOREM 2. Let E be a class of pure exchange economies whose elements e satisfy conditions (a), (b), (c) of Theorem 1, as well as the two following ones:
(d’) zfE containsZ = ((~3, R4))fPNand e’E E,(G) where (I, = (&,..., D), then ~‘EE; (e) for every e E E, e = (el,..., en), ei = (d, R+) for each i E N, the preference relation Ri is convex and continuous7 Furthermore, let the outcomefunction have the foliowing properties:
(i) for each e EE, B++ n &de) C W;
(iv) for each i E N and each rnn( E.A+(, the set Hi(mli)) is convex. Then, for every e E E, B,, n &(e)
c w(e).
[Here B,, = {z: z = (zl,..., zn), zi > -4” for all i E N} is the interior of the Edgeworth Box.] Proof. Suppose the theorem to be false. Then there is an economy Z E E and an allocation z* such that z* E B,, n .A’@) and z* E S(Z) but z* 4 W(.Z). (See Fig. 3.) We shall show that this supposition implies the existence of an economy e, E EL(W) C E such that z* E J$(eL) while z* 6 S(e,). This violation of(i) completes the proof of the theorem. Write, for each i E N, ci
=
jzi:
wi
+
zij&ji
+
z*d).
7 Ri is conuex if,for any a, 6, c E Iw+* and any 0 < 8< 1, c = BQ + (1 - 0)6, the relation a P*b implies cP’b. (xFy means xR’y but not yRk.) R’ is continuous if, for every b E [w+I, the contour sets {a: nR’b} and {c : bR’c} are closed.
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is placed on the inside of the boundary of the, set FIG. 3. Shading such as \ H’(~z*)~‘). Arrows attached to indifference curyes show the direction of preference, A symbol such as I? shows that the indifference curve so labeled corresponds to the preference relation RI.
Denote by m* a Nash Equilibrium m* E v&?)
message generating z*, i.e., such that and
z* = II(
Also, for each i E N, write
Hi is convex by (iv) and Ci by condition (e). Relative to the smallest linear manifold containing the set of all feasible allocations (the Edgeworth Box) both sets have non-empty interiors. (For C”, by condition (c.3); for Hi, by construction.) Furthermore, the monotonicity of preferences (c.3) implies that points of the relative interior of C” are strictly preferred (according to the preference relation JP) to z *i. It follows that the relative interiors of C* and Ht are disjoint and so there exists a separating hyperplane Li. Since Zig belongs to both sets, it also lies in L”. For each i f N, we shall write L’ = {&+ qi * zi = q2 - z**>,
152
LEONID
HURWICZ
and assume (without loss of generality) that qi
. 9
2
qi
. z*i
for all
zi E Ci,
qi
. zi
<
@ . z*i
for all
zi E Hi.
Since Li is a supporting hyperplane for Ci, the monotonicity implies qi E R\+ for all i E N.
condition (c.3)
Suppose now that, for some r, s E iV, q+ # qs. Consider then the economy e’ = ((~9, ai)),,, where, for each i E N, the preference relation I? is generated by the utility function vi(zi)
=
qi
. zim
By (d’), the economy Z belongs to E. Also, because the indifference sets are hyperplanes parallel to Hi, z* is a Nash allocation for t?. However, since
i
22
x2<
I
Y Y2
01.
FIG. 4. Shading such as ‘tc_ is placed on the inside of the boundary of the set We’). Arrows attached to indifference curves show the direction of preference. A symbol such as R1 shows that the indifference curve so labeled corresponds to the preference relation IF.
ALLOCATIONS A'ITAINABLE THROUGH NASH EQUILIBRIA
153
individuals r and s have different marginal rates of substitution at z* which is in the interior of the Edgeworth Box, z* is not Pareto-optimal for 6 It follows that all the q” must be equal, i.e., ql
=
q2
=
. ..
= q” = q (say).
Consider now the economy e, = ((69, RLi)hEN where, for each i E N, the preference relation RL is generated by the utility function vyzq = q * 9. Again, by (d’) e, belongs to E and z* is a Nash allocation for e, (using the Nash Equilibrium message n-tuple m*). (See Fig. 4.) Hence, by (i), z* is individually rational for e, , i.e., q - (t.2 + z*i) > q * ci9
for all j E N.
(1)
Now the vector q defines at z* what Debreu calls “an equilibrium (for Z) relative to the price system q.” Suppose that equality holds for all j E N in (1) above. Then (q, z*) constitutes a Walrasian Equilibrium for Z, contrary to the initial hypothesis. Hence one of the inequalities in (1) must be strict, say q - (63 + z*i> > q * 69. (2) Summing
over j in (1) and taking (2) into account we get
4’ 1’cEN z*j >o which contradicts the feasibility requirement z*j = 0. Hence z* is infeasible and so not Pareto-optimal, contrary to the initial hypothesis. This contradiction completes, the proof of the Theorem. CjaN
COROLLARY TO THEOREM 2.
Replace condition (iv) by the following:
(iv’) for each i E N and each ml”’ E .A@(, hi(mi , m)(c) is a concave function of mi (with m)i(fixed). Then, with all other assumptions of Theorem 2 maintained, remains valid.
its assertion
Proof. It is sufbcient to show that (iv’) implies (iv). But this follows from the fact that iff: X -+ Y is a concave function on a convex subset of a vector space into a vector space Y, then the set F={yEY:y is convex.
154
LEONID HURWICZ
To see that this is so, consider x’, x” E X and
[al Y’ G .fW,
Y”
Also, let 7 = ey’ + (1 - e) y”,
o
Now, by concavity ofS, bl
ef(x’) + (1 - 6 f(x?
G f(x)
for x = 8x’ + (1 - 8) X”.
Also, by [al, [CI w + (1 - ew G efw
+ (1
0)f (x3.
Then fib] and [cc]yield ef + (1 - 0) Y” ~fm, i.e., J Gfc3. Hence F is convex.
2. ECONOMIES WITH PUBLIC GOODS Theorems 1 and 2, as well as the Corollary, stated above for pure exchange economies, have their counterparts for economies in which public and private goods are present and production is possible. In this section we indicate the modifications needed in the above proofs to obtain these counterpart theorems. In a slight change of notation, let x denote the vector of private goods and y the vector of public goods, both of one or more dimensions; both symbols represent, as before, net increments rather than total amounts. Where convenient, and without loss of generality, the first component of x will be used as numeraire. In the present notation, a preference relation R belonging to the class 93, is generated by a utility function of the form 4x9.Y)
=
wr,
+
+
e [ jz
Xl
+
(1
rj W-h,
-
eQ
+
do,,
xj
+
+
tj>
+
xi)
1
+
qs No,. .3
B * Y]
+
ys
+
kd]
ALLOCATIONS
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155
where the index j runs over all private goods other than x1 and the index s runs over all public goods. In appropriate contexts, the x’s and related parameters have superscripts pertaining to individuals holding those goods; so do the parameters related to y’s, but the y’s-as public goods-have no such superscripts. As for production, it will be assumed to be characterized by constant returns. Let T denote the set of possible (x, y) production points. It is then assumed that T is a cone. I.e., ‘if (x’, y’) E T and h > 0, then (k’, Ay’) E T also. An element of Tcan also be written as (&., xi, y) since the x-component is aggregated from all individuals. The simplest familiar example, where x and y are each one-dimensional, is characterized by the production relation C&N xi + y = 0; in this case t i = -xi is the tax (contribution) paid by the i-th individual toward the production of y. The quantity-price combination ((x*~)~.~ , y*, p$, ( P$~)~~~-) is called a Linduhl Equilibrium for the economy e = ((wi, Ri)iaN , T) if (1) the quantity vector is feasible with respect to (w, T); (2) for each i E IV, Zig = (x*~, y) maximizes i’s satisfaction for preference Ri over the non-negative orthant of consumption subject to the budget equality p$ - xi + pzi . y = 0; (3) p$ * x*+p;*y* >P,* * x + Pz . y for all (x, y) E T, where we write x* = c xi and P,* = c pUi. ioN
iEN
(Since we shall be assuming strictly monotone preferences with respect to all goods, there is no loss in defining the budget constraint as equality. Also as mentioned above, the first private good will serve as numeraire, so that Pz, = 1.)
We may note that, because T has been assumed to be a cone, (3) in the preceding definition implies P,* . x*+p;*y*=o.
This is the familiar zero-profit condition under constant returns. Z(e) will denote the set of all Lindahl Equilibria for e; L(e) will denote the set of all Lindahl allocations for e. Bearing in mind the new interpretation of 92, , we can now state the counterpart, for public goods-constant returns economies of the type just described, of Theorem 1. THEOREM 3. Let E be a class of economies with public goods and constant returns technology T whose elements satisfy, mutatis mutandis, the conditions (a)-(d) of Theorem 1 and whose outcome function h satisfies conditions (i)-(iii) of Theorem 1. Then
-44
2 L(e)
for all
e E E.
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LEONID
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ProoJ: The proof follows precisely the outlines of the proof of Theorem 1 and need not be repeated. It is sufficient to note that since, in that proof, we look at the situation of each agent separately, it makes no difference that the prices for public goods are “personalized.” It may also be noted that condition (c.1) of Theorem 1 could be weakened to permit zero initial endowments in the public goods. We now present the counterpart of Theorem 2. THEOREM 4. Let E be a class of economies with public goods and constant returns technology T whose elements satisfy, mutatis mutandis, the conditions (a), (b), (c), (d’), (e), (i), (iv) of Theorem 2. Then
B,, n &‘jj(e) C L(e)
for all
e E E.
Proof. Here again, the proof is parallel to that of Theorem 2. However, there is a divergence due to the different form of Pareto-optimality conditions when public goods are present. We shall therefore set out this phase of the argument in detail. We first proceed as in the proof of Theorem 2 to establish the existence, for each i E M, of a hyperplane Li separating Hi from Ci. Write pi = {(xi, y): ai . X’ + bi * y = ai ex*~
+ bi * y*}-
We shall normalize so that the first component of ai equals 1. Again, we consider the auxiliary economy S?where all preferences are generated by linear utility functions, with indifference sets which are hyperplanes parallel to L” for each i. As in the proof of Theorem 2, the initially given allocation z* is Nash for the economy E. It follows that a, = a, = -*- = a, = a (say),
since otherwise there would be unequal marginal rates of substitution at an interior allocation and this would contradict Pareto optimality of z* for the economy e’. Consider now the economy eL where, for each i E N, the preference relation RLi is generated by the utility function V'(Z)
Since z* is individually
= a * xi + bi *y,
rational, we have
a . x*( + bi - v* > 0
for all
i E N.
But if equality were to hold for all i E N in (l), z* would be a Lindahl
(1)
allo-
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cation. Hence the left-hand side in (1) must be positive for at least one individual. But then, summing over all individuals, we obtain a*X*+B.y*
>O
(2)
where X* = 1 x*i and B = 1 bi . ieN
iEN
However, we shall now show that (2) implies the non-optimality of z* for e, . This completes the proof since z* is Nash for e, . To show that (2) implies non-optimality, we construct an allocation that is Pareto-superior to z*. This allocation z** = ((x**i)iGN, y**) is defined as follows: y** = (1 + 6) y* where 6’0; x**j = x+-j forj = l,..., n - 1 where x#j is defined below; X**n
=
x**
_
12-l
n-1
cx**j=(l+e)x*j=l
i=l
Cx#j=(14).*--..+,#,
and x**
=
1
x**~
= (1 + 0) X* and X# = c J+.
EN
GN
In the preceding formulae, we define, for each i E N, the vectors x#{ by the equality (3)
It follows that z** is technologically feasible and that, except for individual n, everyone remains at the z* level of satisfaction. It is therefore sufficient to show that the n-th level of satisfaction is higher at z** than at z*, since this implies the non-optimality of z*. Now z**” is preferred by n to Zig if a*x**la-j-b;y**
>a*x#n$b;y**
which, after some cancellations, is seen to be equivalent to a . [(l +
e)X*
- P]
> 0.
(4)
On the other hand, summing (3) yields a . X# + B - ey* = a - x*.
(5)
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LEONID HLJRWICZ
Solving for B * y* from (5) and substituting
into (2) then gives
a - x* + (l/@( x* - aX#) > 0.
(6)
(The division by 8 is legitimate since d is assumed positive.) Multiplying (6) by positive 0 does not change the direction of inequality. Hence,
which is the same as (4). The proof is complete. COROLLARY TO THEOREM 4. Finally, and without Q formal statement, we note that there is a Corollary ot Theorem 4 which is the exact counterpart of that for Theorem 2, with the identical proof.
3. ECONOMIES WITH PRODUCTION In this section we show that the results of Section 1 (where pure exchange was assumed and public goods were absent) can be extended to economies with production (but without public goods). A further extension to economies with production and public goods is straightforward, (Indeed, the model of Section 2 does involve production but under the special assumption of constant returns.) To describe the environment involving production, we write Tj to denote the production set of the j-th producer, j E J. An element tj of Ti, called a (net) output vector, has positive components for outputs, negative ones for inputs. An environment is then defined as e = ((wi, R&+ , (Tj)jsJ). An allocation s = ((z~)~~~, (tj)& is defined in terms of the net increments zi for consumers and output vectors ti for producers. Feasibility requires that the allocation be balanced, i.e., that
and, also, that each CJ + zi be in the individual consumption set (here assumed to be the non-negative orthant IJ!+~ of the commodity space). The n-tuple (&EN is called a distribution. An allocation s is, as usual, called Pareto-optimal if it is feasible and if there is no other feasible allocation whose distribution dominates that of s in terms of preferences. The set of allocations Pareto-optimal for e is denoted by ~(4. An I-tuple p* of reals is called an eficiency price (uector)fir the environment e at the allocation s* = ((z*~)~~~, (t*j)j,J) if s* is Pareto-optimal for e and if: (1) for each i E N, z*( maximizes Ri-satisfaction subject to the budget
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constraint p* * 9 < p* * z*i, and (2) t*j maximizes (the efficiency profit) p* . tj for tj in Tj. [In Debreu’s terminology [2, pg. 931, s* is an equilibrium relative to the price system p*.] It is known that under certain assumptions of convexity, continuity, etc., every Pareto-optimal allocation s has an efficiency price associated with it; indeed, there may be more than one such price if the environment e is not “smooth.” We shall therefore find it convenient to introduce a function # selecting a particular efficiency price Jl(e, s) from the set (assumed non-empty) of the efficiency prices for e at s. The price (vector) #(e, S) will be called the selected eficiency price for e at s. It is perhaps not quite obvious what the appropriate generalization of the individual rationality concept should be. It is natural to seek a distribution (called the reference distribution) which would play here a role analogous to that played by the initial endowment n-tuple (ui)ieN in the case of pure exchange, as well as in the case of constant returns. (This approach is in line with the framework introduced in [Hurwicz, 4, pp. 18-191. A private communication by William Thomson concerning the analogues of Theorems 1 and 2 for “envy-free” (equitable) distributions was extremely helpful here.) Clearly, the reference distribution will depend on the environment. But in the case of production, it will also depend on how much was produced, by whom, and, perhaps other factors. We therefore introduce a function y, called the guarantee structure specifying the reference distribution given the environment e and the allocation s. If z* = (z*~)~~~ is the reference distribution generated by the guarantee structure y, we write z* = r(e, s), or z*( = yi(e, s), i E N. An allocation s# = ((z+QN , (t#&) is said to be individually rational with respect to the guarantee structure y for the environment e if and only if, for
where ki is the i-th preference relation (Ri) in e. Of particular interest is the$xed share guarantee strucfure where r(e, s) is given by f(e, s) = wi [ 1 + (P * ; e’(fj)/(p
. Ws] , Z-EN,
(1)
where p = #(e, s) is the selected efficiency price for e at s and the eji are nonnegative fractions such that xi 8{ = 1 for each j E J. We shall denote by $Je) the set of allocations that are individually rational with respect to the guarantee structure y for the environment e, and, as previously, we shall write S,(e) = g(e) n YJe). In particular, S,(e) and S,(e) will denote the corresponding sets with respect to the guarantee structure specified by (1). 642/21/1-11
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The game is again defined by the message (strategy) spaces JP and the outcome function h. However, here the outcome function must specify outputs as well as the distribution. We write, therefore, h = ((hri)ienr , (h&,) where hGi specifies zi and hf specifies ti, in both cases as functions of the n-tuple (m, ,..., m,), mi E .Mi for each i E N. As in the introductory section, we define, for each i E N and each rnji( E dPc ht(dP,
m)i() = (zi: zi = hzi(mi , rn)Q) for some mi E A?),
Hzi(m)“() = {zi: .zi < hzi(mi , m)i() for some mi E Ai}. Next, the set of Nash Equilibrium
strategies for the environment
e is defined as
vh(e) = {m*: for all i E N, hzi(m*) Rihsi(mi , m*ji() for all mi E Aft}, and the set of Nash allocations as A$(e) = {s: s = h(m) for some m E vh(e)}. We also need unambiguous notation for the familiar concepts related to the competitive equilibria of a private ownership economy in which the i-th consumer owns the share eji of the j-th production unit and is, consequently entitled to the corresponding fraction of its profits; here the i-th budget constraint has the form p’Zi
Although this formulation is due to Arrow and Debreu [l, pp. 270-2711, we shall give priority to considerations of uniformity in notation and terminology, and continue to refer to competitive equilibria and allocations as Walrasian. We shall denote by ?‘KO(e) the set of price-allocation pairs (p, S) constituting competitive equilibria for the environment e given the ownership structure specified by the matrix 19= (eji). The corresponding set of competitive allocations will be denoted by W,(e). Before stating, as Theorem l*, the counterpart for economies with production of Theorem 1, we need one more notational convention. In Theorem 1, we used the symbol EL,-(w) to denote the class of environments with the initial endowment n-tuple w and preferences from 9, or 9?L . In Theorem 1* and its proof, on the other hand, we shall use in corresponding places the symbol E&w, T) denoting a class of environments having the following property: Let e = ((wi, Ri)islv , (T&) be a given environment; let (tj)jEJ be a set of output vectors such that, for each j E J, the output vector tj belongs to Tj; let the vector pj define a hyperplane supporting Ti at tj; and, for each i E N, let RiEg’,Ug’,. Then the class of environments EL&w, T) contains an
ALLOCATIONS
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environment 8 = ((09, IP)i,, , (pj)j.J) such that, for eachj E J, tj is the unique point common to Ti and the hyperplane defined by pi. We may note that if Ti is “strictly convex at tj,” it then qualifies asr?‘i in the preceding definition. (We do not require that Tj be convex, but the requirement of the definition is vacuous unless Ti has a supporting hyperplane at tj.) Analogously, we define a class Et(w, T) of environments. Its definition differs from that of E&w, T) only in that the preferences all must come from B’r. , rather than from Br. u B’c. Let E be a class of economies e = ((CC/,Ri)i4N, (Tj>& with production satisfying the conditions of Theorem 1, with (d) replaced by (d*) in which E&w, T) takes the place of E&o). Furthermore let the outcome function h (as defined earlier in the present section) have the following properties: THEOREM
(i*)
1 *.
for each e E E,
(ii*) and (ii*), where the latter two conditions are obtainedfrom (ii) and (iii) of Theorem 1 by having ELc(u, T) and E,(w, T) take the respective places of E&w) and EL(w). Then
Proof. Since the proof is quite similar to that of Theorem 1, it will only be sketched here. Let e E E and d E W,(Z)). We shall show that SE Jr/-,(Z). First, we choose - _ price vector p so that ( p, s) E Y&(Z) and p = #(a, S); i.e., j is the selected efficiency price for Z for S. (Since dlrg(?) is non-empty by hypothesis, the price p is well-defined.) Now construct a sequence of economies e, from E&6, T), converging to some eL in EL(6, T), as follows. Choose preferences $” for e, just in the proof of Theorem 1 except that & must be replaced by the commodity bundle
where P is the j-th output component of S. As for technologies, let Fj be so chosen that pi defines a hyperplane supporting T’j at P, and ii is the unique point common to this hyperplane and T*. Then let, for each j E J and all v = 1, 2,..., Tvj = Td, and e, = @“, R%,N 9 (c%J).
162
LEONID
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Given this construction it can be shown that the diameter of &(e,) + 0 as v --f co, and SE &(e,) for all v. By our assumptions, each e, has a Nash allocation S, and S, + S. Hence the closed graph property implies that s E M,,(e,) where eL = lim, e, and eL E EJw, T). But then, by reasoning analogous to that used in the proof of Theorem 1, it follows that s E .AqZ). Before stating, as Theorem 2*, the counterpart for economies with production of Theorem 2, we introduce a class of environments denoted by E,*(w, 7). Given an environment e = ((wi, R3rEN , (T&), this class consists of all the environments of the form d = ((ai, ai), (Tj)j,J) where Ri E 9ZL . I.e., it contains all environments with the same initial endowments and technologies as e while preferences are generated by linear utility functions. THEOREM 2 *. Let E be a class of economies with production s’atisfying the conditions of Theorem 2, with (d’) replaced by (d’*), using Ez(w, T) in place of EL(w), and the additional condition: (f) for all j E J, the set Tj is convex. Furthermore, let the outcome function h satisfy: (i*) for each e E E, B,, n &te) C X44; (iv*) f or each i E N and each rnji( E .A%‘)~(,the set Hzi(mji() is convex. Then, for every e E E, B,, n 4$(e) C W,(e).
Proof. Here again the proof follows the pattern of that for Theorem I, again with the right-hand side of (1) replacing wi as the “guaranteed” commodity bundle. It should only be noted that the common value q of the qi (“slopes of the separating planes Li) in that proof constitutes the selected efficiency price for the given Pareto-optimal allocation s* because eL has smooth (in fact, linear) preferences and so there is only one efficiency price at s* which must coincide with q. Thus let s* = (z*, t*) belong to X*(Z) and to S,(?) but not to W,(Z). We show that this yields a contradiction. For each i E N, let Li = {zi: qi . zi = qi . Zig} be the planes separating the sets Hi(m*)i() from the set Ci. Let e’ be the economy with the same initial endowments and technologies as 2 but with the i-th preference relation generated by vi@) = qi * z”. Again s* is a Nash allocation for ?, and so must be Pareto-optimal. This implies that all the qt must have a common value, say q. In turn, we consider the economy eL with initial endowments and technologies same as in Z but preferences generated by vi(zi) = q . zi. Again, s* is Pareto-optimal for eL and q is its (here unique) efficiency price. The requirement of individual rationality with respect to the fixed share guarantee structure 0 implies that
vyz**) > vi [Wi(q - T B,“t*j)/(q *cc+)]
for each i E N,
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i.e., that q . i!*i 2 q * 09 (q * c Bj’t*j)/(q
. C&)
for each i E N.
j
(2)
But equality cannot hold in (2) for all i f N for then (q, s*) would constitute a Walrasian Equilibrium for 2, contrary to the initial supposition. (It is obvious the budget equalities would be satisfied in virtue of (2) and that consumer satisfactions would be maximized because of the separating plane property of q. But profit would also be maximized by the efficiency price property of 4.1 Hence for at least one i’ E N, the inequality in (2) must be strict. It follows that when both sides of (2) are summed over i, the left-hand sum will be greater than the right-hand sum. Thus we obtain (after cancelling the term q in the right-hand side of (2) and summing over i)
q’ci
z*i > q . c c e,w z j
Now -& xj e,it*i = Cj t*j xi Bji = Cj t*j. Therefore (3) becomes
4.F
z*i > q . c t*j, i
(4)
which contradicts the feasibility (balance) condition x:f z** = Cj t*j. Since the latter must be satisfied by the Pareto-optimal allocation s*, a contradiction has been arrived at and the proof is complete. Remarks 1. We may note that, since “efficiency profit” p * tj = 0 under constant returns, the case where constant returns prevail retains the initial endowment as the “guaranteed” commodity bundle under the fixed share structure. Therefore, as in Section 2, production with constant returns does not require the more complex machinery of “guarantee structures.” 2. It appears that guarantee structures other than fixed shares could be handled by analogous methods. However, they would require appropriately modified formulae for sharing profits in a private ownership economy.
4. AN EXTENSION
TO NON-PLAYER
PARTICIPANTS
(QUASI-GAMES)
From the point of view of economic interpretation the preceding game structure is somewhat inadequate. In the presence of production it is natural to introduce information-processing (decision-making) entities (agents) who obey certain behavior rules but do not have preferences, and hence are not players in a game (“soulless” production managers). Even in the pure exchange case economists often find it convenient to introduce an auctioneer
164
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who is not a player. Of course, one may wish to endow the auctioneers or the production managers with preferences so as to take better account of incentive issues, but it is desirable to have room for non-player participants. This can be accomplished by defining what we shall call a quasi-game G. In this quasi-game there is a set of players N each of whom controls a strategy variable mi and a set of non-players K each of whom controls a decision variable dk in DL. The outcome function h associates an allocation as a function of the mi and the dk , say s = h((m& , (d,J&. Each player has a preference relation Ri on his component of the distribution in s. A non-player, by contrast, has only a behavior rule Fk which specifies the permissible values dk E DL given the strategies mi and the decisions d,f ED”‘, k’ E K - jk), and, possibly, also some information concerning the environment. (For instance, no such information is ordinarily assumed be used by an auctioneer, but it would be assumed known to a non-player manager of a production unit.) We express this by dk E Fk(m, dbk(, e)
k E K.
Now a Nash Equilibrium of the quasi-game G is defined as follows: given the strategies of other players and the decisions of the non-players, no player can secure a preferred outcome by modifying his own strategy; given the strategies of all the players and the decisions of other non-players, each non-player’s decision is permissible for the prevailing environment. Formally, (m*, d*) is a Nash Equilibrium of G for e if and only if; (1) for each i E N, hzi(m*, d*) Rihzi(mi , m*)i() for all mi E J@, and (2) for each k E K, d*” E Fk(m*, d*jk(, e).
A cursory examination indicates that our results can easily be extended to quasi-games of the type just described.
REFERENCES I K. J. ARROW AND G, DEBREU, Existence of an equilibrium for a competitive economy, Econometrica 22 (1954), 265-290. 2. G. DEBREU, “Theory of Value,” Wiley, New York, 1959. 3. T. GROVES AND J. LEDYARD, Optimal allocation of public goods: A solution to the “free rider” problem, Econometrica 45 (1977) 783-809. 4. L. HURWICZ, The design of mechanisms for resource allocation, Amer. Econ. Reo. 63 (1973),
l-30.
5. L. HURWICZ, Outcome functions yielding Walrasian and Lindhal allocations at Nash equilibrium points, mimeographed handout for presentation at a Stanford Seminar, November 21, 1976. (An earlier version circulated for Econ 208, Berkeley.) Reo. Econ. Studies, in press. 6. L. HUR~ICZ, Balanced outcome functions yielding Walrasian and Lindhal allocations at Nash equilibrium points for two or more agents, Boulder, Cola., September 3, 1977
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(private circulation). Presented at the Econometric Society meetings, Boulder, Colo., June 1978. 7. K. MOUNT AND S. REITER, The informational size of message spaces, J. &on. Theory 8 (1974), 161-191. 8. S. REITER, Information, incentive, and performance in the (new)z welfare economics, Amer. Econ. Rev. 67 (1977), 226-234. 9. D. SCHME~LER, A remark on microeconomic models of an economy and on a gametheoretic interpretation of Walras equilibria, mimeographed handout, Minneapolis, Minn., March 1976.
OF ECONOMIC
THEORY
On Allocations
21, 140-165 (1979)
Attainable
through
Nash Equilibria*
LEONID HURWICZ+ Department of Economics, University of Minnesota, Minneapolis, Minnesota 55455 Received July 28, 1977; revised December 14, 1978
The purpose of this paper is to show that there are severe limitations on allocations attainable through mechanisms which can be viewed as Nash noncooperative games in which the agents’ messages constitute their strategies. In such mechanisms, the designer chooses the individual agents’ message sets and the rule (called outcomefunctionl) specifying the resource allocation resulting from the agents’ message choices. The messages chosen by agents constitute a Nash (noncooperative) equilibrium if no agent can unilaterally improve his situation as long as others do not change their messages. Clearly the equilibrium messages will depend on the agents’ characteristics (initial endowments, preferences, technology) and, hence, so will the allocations corresponding to these messages. We shall refer to the totality of all the agents’ characteristics as the environment and we shall call the allocations generated by Nash equilibrium messages Nash allocations. The relationship between environments and the corresponding Nash allocation sets specified by an outcome function constitutes the (Nash) performance correspondenceaassociated with that outcome function. The designer of the mechanism is assumed to be interested in the performance correspondence, but he can only control it indirectly by choosing the outcome function and the message sets. This paper shows that such indirect control is quite limited: only certain types of performance correspondences can be generated by a Nash equilibrium mechanism even if the designer has great freedom in selecting outcome functions and message sets. The economist’s interest in the design of resource allocating mechanisms * In addition to a general intellectual indebtedness in this area of analysis to T. Groves, J. Ledyard, S. Reiter, and D. Schmeidler, I also want to acknowledge helpful conversations with .I. Jordan (who pointed out an error in an earlier version of this paper), A. Postlewaite, R. Radner, L. Shapiro, S. Smale, and W. Thomson, The appointment as visiting Ford professor in the Economics Department at Berkeley (U. of Calif.) provided an ideal setting for this research, with much of the stimulus due to M. Kun and the summer IMSSS seminars at Stanford. + Project supported by NSF Grants GS-31276X’and SOC 76-14786. 1 Also called “game form” by other writers. a We follow here the formulation due to Mount and Reiter [7]. “Social choice correspondence” is another term used.
140 0022-0531/79/040140-26$02.00/O Copyright All rights
0 1979 by Academic Press, Inc. of reproduction in any form reserved.
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was greatly stimulated by the “free rider” problem arising in connection with public goods. In line with the prevailing tradition, interest in this area was focused on Pareto-optimality. Groves and Ledyard [3], whose model played a crucial role in stimulating the present line of research, constructed outcome functions whose Nash allocations were always optimal, but not necessarily individually rational; i.e., there is no guarantee that the equilibrium satisfaction levels of agents are at least as high as those at initial endowments. It became natural to inquire whether a different choice of outcome functions would yield a performance both Pareto-optimal and individually rational for all economies of a specified class. One way to accomplish this was to find outcome functions yielding the Lindahl performance, since the latter is both Pareto-optimal and individually rational. Such outcome functions, both balanced and not balanced, were shown to exist in Hurwicz [5] for economies involving both public and private goods, where the latter can be either consumed or used as inputs to produce public goods under constant returns. However, the Lindahl correspondence is only one among many that guarantee Pareto-optimal and individually rational allocations. It is therefore natural to ask whether other correspondences with the two attributes of Pareto-optimality and individual rationality can also be realized through Nash equilibria. Section 2 of the present note3 provides a partial answer to this question. Theorem 3 shows that, under certain assumptions on the environment class covered, and assuming a continuity property for the performance correspondence, if all Nash allocations for a given environment are Pareto-optimal and individually rational, then every Lindahl allocation for that environment is among its Nash allocations. Theorem 4 constitutes a partial converse. It states that, again under certain assumptions on the environment class covered, and assuming a convexity property for the outcome functions, every interior Nash allocation is a Lindahl allocation for the given environment. The nature of the convexity property of the outcome functions is exemplified in the Corollary4 to Theorem 4 which implies that the following is suficient: for each participant, the outcome function is concave in this individual’s own message (strategy). However, the convexity condition of Theorem 4 is significantly weaker. Of course, in Theorem 4 and its Corollary it is again postulated that every Nash allocation is Pareto-optimal and individually rational for the given environment. 8 Although historically the issues of this paper arose primarily in connection with public goods, we find it convenient in the body of this paper to reverse the order and to start with private goods (“Edgeworth Box”) economies. ‘The Corollary to Theorem 4 is an exact analogue of the Corollary to Theorem 2, with an identical proof. Only the Corollary to Theorem 2 and its proof are presented formally.
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Assumptions on the environment are of two types. On the one hand, they require that every economy belonging to the class covered have certain properties. On the other hand, they require that the class covered be sufficiently rich; in particular, that it contain certain special linear and strictly concave economies described at the end of the section on notation. As for requirements of the first type, both theorems assume nonnegative orthant as consumption set and strictly positive initial endowments, as well as preferences strictly increasing in every good. In addition, Theorem 4 and the Corollary postulate continuous convex preferences and refer to interior allocations only. It may be noted that the assumptions on the performance correspondence, including the continuity property (closed graph) mentioned above, are satisfied by the Lindahl correspondence. Hence our results are not vacuous. On the other hand, one can easily think of reasonable performance correspondences that are excluded by these results. One example is given by a performance correspondence whose allocations maximize the product of utility gains from the initial level (the Nash bargaining solution). Another, by a performance correspondence whose allocations maximize utility gains subject to the constraint that these gains be equal for all participants (or satisfy some other a priori imposed ratio requirements). In the latter two examples it is to be understood that some canonical utility indices have been specified.5 It appears that the Groves-Ledyard outcome functions satisfy the hypotheses of our two Theorems. Since they do not yield the Lindahl correspondence and do guarantee Pareto-optimality, it is not surprising that individual rationality is violated. Analogous results are obtained for economies in which public goods are absent. In Section 1, we consider pure exchange economies with private goods only, For such economies, outcome functions realizing the Walrasian performance correspondence through Nash Equilibria were first constructed by Schmeidler [9] and later Hurwicz [5, 61. Again, the question arises what other performance correspondences, guaranteeing Pareto-optimality and individual rationality, could be realized through Nash Equilibria. For cases so far investigated, the answers are analogous to those obtained for the public goods model. In particular, Theorem 1 shows that, given a continuity property of the performance correspondence, and certain assumptions on the class of environments covered, if all Nash allocations for a given environment are Pareto-optimal and individually rational, then every Walrasian allocation for that environment is a Nash allocation. Theorem 2 again constitutes a partial converse of Theorem 2. Under 5 For instance, in a classical environment let each agent’s utility of a given bundle z’ be measured by the distance from the origin 0 to the point z” which is on the same indifference surface as z’ and which contains the same quantities of all goods; i.e., z” -i z’, z = (z; ,..., z;,, z: = 1.. = z; , and u*(z’) = ((z;)~ + ... + (z;)~)~/~.
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assumptions analogous to those of Theorem 4, every interior Nash Equilibrium allocation is Walrasian. The Corollary to Theorem 2 shows that the convexity condition in Theorem 2 is satisfied when every individual’s outcome function is concave in this individual’s message (strategy). However, the convexity condition of Theorem 2 requires much less than the concavity of outcome functions. In Section 3 we extend the results of Section 1 to economies with production although without public goods. (A further extension to economies with production and public goods is possible.) Theorems 1” and 2* are the respective counterparts of Theorems 1 and 2. It should be noted that when production sets are convex but constants returns do not prevail everywhere, the appearance of “profits” necessitates ‘a reconsideration of the notion of individual rationality. This is so because it would not in general make sense to use the initial endowment as the reference point for utility comparisons. We deal with this problem by introducing the notions of a reference distribution and a guarantee structure, and speak of allocations that are individually rational with respect to specific guarantee structures. One such structure is that of fixed shares (defined in (1) of Section 3) which is analogous to a “private ownership economy” of the Arrow-Debreu type. Alternative guarantee structures would also deserve investigation, but this has not been done in the present note. It is rather remarkable that our results are valid whether or not the outcome functions are balanced away from equilibrium. Furthermore, they are valid not only for games proper but also for what we call quasi-games in which non-players may also participate. This allows for the introduction of auctioneers and “souless” production managers who do not have preferences but have decision rules. The extension of our results to quasi-games is discussed in Section 4. NOTATION
R R+ Iw++ W (resp. R+*, R:+)
The The The The
set of set of set of q-fold
all reals all nonnegative reals all positive reals Cartesian product of R (resp. of R+q, Rt+)
Environment n
N = (l,..., n} 1 6Ji Ri or &
642/2III-IO
The number of agents (n > 2) The set of agents The number of goods (I > 2) The i-th initial endowment; w = (wl,..., w-) (wi E R:+) The i-th preference relation (reflexive); R = (Rl,..., R”); (Ri C (w+I x [w+I)
144
LEONID
Pi or >$ -i ed = (cd, Ri)
HURWICZ
The i-th strict preference relation The i-th indifference relation The i-th characteristic; e = (el,..., e”) an environment (an economy) A class of pure exchange economies e with [w,’ as consumption sets
E Allocations zi
Net increment in commodity holdings by individual i (zi E W) An allocation The set of allocations Pareto-optimal for e The set of allocations “individually rational” for e [i.e., allocations z such that, for each i E N, wi + ziRiwi where e = ((wi, Ri))i6N]
2 = (z’,..., z”) 9(e) C Rzn X(e) C Rzn
S(e) = .9(e) n 9(e) W(e) E Rz+ln
The set of pairs (p, z) constituting Walrasian Equilibria for e (p * zi = 0 for each i E N) The set of allocations Walrasian for e; W(e) = {z: (p, z) E W(e) for some p}
We> The Game &vi JpiC
The i-th message (strategy) space =A1
x
h”: ud! -+ [wz
.. . x
&i-l
x
di+l
x
. .. x
4%
The i-th outcome function (its values are net increments in commodity holdings by i-th individual) we write zi = hi(mi , m)i() where mi E &Ii and m’i’ g &?pc The outcome function h = (hl,..., h”) hi(Mi, rnji() = {zi: zi = hi(mi , rnji’) for some mi E &!i} where rnji( is a fixed element of JZ)ic Hi(m) = {zi: zi < hi(mi , rnji() for some mi E d@>, rn)i’ E JY)Q vh(e) = {m*: Vi E N, oi + h”(m*) Riwi + hi(mi , m*ji() for all mi E Xi} where e = ((wi, Ri))ieN = the set of Nash Equilibrium message n-tuples for the economy e when the outcome function h is used v,,:E+.& The Nash Equilibrium message correspondence &(e) = {(zl,..., zn) E !Rzn: Vi E N, zi = hi(m) for some m E vh(e)} = the set of Nash Equilibrium allocations for e when h is used The Nash performance correspondence when h is A$: E -+ (Wzn used Jh = h 0 vh
ALLOCATIONS
ATTAINABLE
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145
Certain Special Environments x IR,.z* R ‘is a preference relation generated by a utility function of the form v(x, y) = x + /? . y for some /3 E lRq;;‘> where x E R and y E Rz-l
@)L=(RCR+Z
ELi(oi) = {ei: ei = (cd, R”), Ri E 9?,.> EL(w) = {(el,..., en): Vi E N, ei E ELi(wi)} 9?c = {R C R+z x R+z: R is a preference relation
generated by a utility function of the form u(x, y) = w. + x + (1 - 0)~ * (oy + y) + with 0<8<1; q$,k,>O for s= 8 CL’, qs ln(w, +yg+kk,) 1,..., 1 - 1; & E [FB:::) E RZ-l where W, E R,, , XE!R,WyER~~,Y
Eci(wi) = (ei: es = (cod,Ri), Ri E a,-} E,(w) = {(el,..., en): Vi E N, ei EEci(w”)} h(w) = W,..., en): Vi E iV, ei E ELi(wi) U Eci(w”)}.
1. PURE EXCHANGE
ECONOMIES WITHOUT
PUBLIC
GOODS
1.1. THEOREM 1. Let E be a class of pure exchange economies whose elementse satisfy the following conditions: (a) n>,2; (b) l 3 2; (c) for each i E iV, (c-1) d E lR;+ , (c.2) the admissibleconsumptionset is Iw+I; (c.3) the preference relation R” is strictly increasing in each good;
(d) ifE contains B = ((&, e))i.N and Z E J?&(G) where (I, = (G,..., 3) then e’E E. Furthermore, let the outcomefunction h have the folIowing properties:
(i) for each e E E, .4(e) C S(e);
(ii)
zye E E,,-(W) C E, then 4(e)
(iii)
# 0;
ifELc(&) C B; e, E EL,-(&), z” E &(e,), IJ = 1, 2 ,...; e. E W-3;
then z, E J$(eo).
e, = lim e, ;
z, = lim z, ;
146
LEONID HURWICZ
Then X$(e) 2 W(e)
for all
e E E.
ProojI Let 2 = ((69, ai)),, belong to E. We shall show that every Walrasian allocation for 2 is also a Nash allocation, i.e., that Z E W(Z) implies P E .4$(Z). Let then ( 3, 2) be a Walrasian equilibrium for 2. Since 62 > 0 by (cl), there is a good j such that ojl + Zjl > 0. We shall denote this j-th good by x and other goods by ys, s = I,..., I- 1, so that 6,’ + X1 > 0. We shall also write zi = (xi, yi) and p = ( & , &,) = (1, &,). Such normalization of prices is legitimate, since, by (c.3), p. > 0. Let r = I,..., IZ - 1, and consider a sequence of characteristics v = 1, 2,...,
eYT = (67, R,?),
where RUTis generated by the utility function6 U”‘(X’, y’) = WzT+ XT + (1 - e,) py . (G*r + y’) Z-1 +
0,
1 s=1
9;
ld4,
+
Y,7
+
k,')
and k,’ > 0 for all s = I,..., I-l;O~&,~1;lim8,=0;and 4.: = PY (6s + js’ + k,‘) for s = l,..., 1 - 1. Using the sequences eVr, r = I,..., n - 1, of characteristics just defined, we now construct a sequence e, = (e,l,..., e:-‘, eLn) of economies where e,” = (w”, RLn) is generated by the utility function v” = (x”, y”) = 8,” + xa + p, * (f&n + y”) = p * (3 + 2”). Observe that lim e,’ = eLr = (G”, R,?), where RLT is generated p * (6~ + z?). Therefore
r = l,..., n - 1,
by the same linear
utility
lim e, = e, = (eL1,..., eLn),
function
v’(z’) = (1)
an economy whose initial endowments are the same as in 2 and whose indifference sets are planes parallel to the Walrasian price (budget) plane through 5. 6 Other types of utility functions could also be used. Slightly different construction may be preferable when f is on the boundary.
ALLOCATIONS
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147
We shall now show that Z is a Nash allocation for the economy eL . In view of the “closed graph” property (iii), it will sufike to find a sequence of allocations z, such that zVis a Nash allocation for e, and lim z, = E
(2)
(See Fig. 1.)
FIG. 1. Arrows attached to indifference curves show the direction of preference. A symbol such as I? shows that the indifference curve so labeled corresponds to the preference relation R’ .
Since each economy ‘e, belongs to the class ELC , condition (ii) guarantees that it does have a Nash allocation. Furthermore, by (i), this allocation is located in the sets S(e,) of Pareto-optimal individually rational allocations. Now the sequence e, has been so constructed that the diameter of S(e,) -+ 0 as v ---f cc
(3)
and Z l S(e,)
for all
Therefore any sequence of allocations satisfy (2).
v = 1, 2,... .
(4)
zV such that z, is Nash for e, must
14%
LEONID
HURWICZ
To see that (3) and (4), hold note first that, for each v, an allocation w, Y%N is Pareto optimal for e, only if y’ = p” for all i E N. This is so because, for v = 1,2 ,... and s = l,..., I - 1,
and
auyay,r Iv+ = py, for r = 2,..., n. It follows that 5 is Pareto-optimal for every e, . As for individual rationality, it is seen that within the Pareto-optimal set @(e,) -where yi = jY for each i E N-we have ~,i(z,t) > u:(O) for all i EN if and only if
x,n a P and for all
5’ 2 5s.’
r = l,..., n - 1
where e is defined by the equality
or, explicitly,
In fact, the above inequalities on the xf together with the conditions yi = .j” for all i E N and the requirements of feasibility (staying in the Edgeworth Box) define the set S(e,) for each v. Since &;’ < Kr for all v, it follows that Z is both Pareto-optimal and individually rational in every economy of the sequence, so that (4) is satisfied. Noting that X* = -& * y’ for all i E N, we see from the formula for 5,’ that .&;’ + XT as v-00, r = I,..., n - 1 since the 0, term drops out asymptotically. Furthermore, since the x:, as well as the + add up to zero, it follows from the inequalities on the xV7that n-1 c #=l
m-1 x,7
<
c r-1
x7.
Using relations involving the 5,’ we conclude that x,’ + Z. This implies that x::; (xy’ - 57) + 0, and hence, x,” -+ Zn, Thus x, -+ Z, and hence z, = (xy , J) + (Z, J) = 1. So (2) has been established. (Of course, (3) is implicit in this argument.)
ALLOCATIONS
ATTAINABLE
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149
EQUILIBRIA
So far we have shown that
This means that, for some m* E .A!, m* E vded
(5’)
Z = h(m*).
(5”)
and
Hence, because the indifference surfaces for each eL* are given by J * 3 = const., the fact that Z is a Nash allocation for eL implies that, for each i E N,
But then (6), in conjunction with (57, implies that m* E ~~(2). For, by hypothesis, (p, 2) is a Walrasian Equilibrium for .Z, and so every trade z” preferred to 3 (according to the preference relation I@) must cost more
\
ol
,5
-. %
FIG. 2. Shading such as d/ is placed on the inside of the boundary of the set /+(A’, m*“‘). Arrows attached to indifference curves show the direction of preference. A symbol such as I? shows that the indifference curve so labeled corresponds to the preference relation RI.
150
LEONID JXJRWICZ
than p - 9. (See Fig. 2.) Hence it cannot be reached by individual of the implication in (6). It follows that, for each i E N,
i in virtue
and, therefore,
This completes the proof of the Theorem. 1.2. THEOREM 2. Let E be a class of pure exchange economies whose elements e satisfy conditions (a), (b), (c) of Theorem 1, as well as the two following ones:
(d’) zfE containsZ = ((~3, R4))fPNand e’E E,(G) where (I, = (&,..., D), then ~‘EE; (e) for every e E E, e = (el,..., en), ei = (d, R+) for each i E N, the preference relation Ri is convex and continuous7 Furthermore, let the outcomefunction have the foliowing properties:
(i) for each e EE, B++ n &de) C W;
(iv) for each i E N and each rnn( E.A+(, the set Hi(mli)) is convex. Then, for every e E E, B,, n &(e)
c w(e).
[Here B,, = {z: z = (zl,..., zn), zi > -4” for all i E N} is the interior of the Edgeworth Box.] Proof. Suppose the theorem to be false. Then there is an economy Z E E and an allocation z* such that z* E B,, n .A’@) and z* E S(Z) but z* 4 W(.Z). (See Fig. 3.) We shall show that this supposition implies the existence of an economy e, E EL(W) C E such that z* E J$(eL) while z* 6 S(e,). This violation of(i) completes the proof of the theorem. Write, for each i E N, ci
=
jzi:
wi
+
zij&ji
+
z*d).
7 Ri is conuex if,for any a, 6, c E Iw+* and any 0 < 8< 1, c = BQ + (1 - 0)6, the relation a P*b implies cP’b. (xFy means xR’y but not yRk.) R’ is continuous if, for every b E [w+I, the contour sets {a: nR’b} and {c : bR’c} are closed.
ALLOCATIONS
ATTAINABLE
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NASH
EQUILIBRIA
151
is placed on the inside of the boundary of the, set FIG. 3. Shading such as \ H’(~z*)~‘). Arrows attached to indifference curyes show the direction of preference, A symbol such as I? shows that the indifference curve so labeled corresponds to the preference relation RI.
Denote by m* a Nash Equilibrium m* E v&?)
message generating z*, i.e., such that and
z* = II(
Also, for each i E N, write
Hi is convex by (iv) and Ci by condition (e). Relative to the smallest linear manifold containing the set of all feasible allocations (the Edgeworth Box) both sets have non-empty interiors. (For C”, by condition (c.3); for Hi, by construction.) Furthermore, the monotonicity of preferences (c.3) implies that points of the relative interior of C” are strictly preferred (according to the preference relation JP) to z *i. It follows that the relative interiors of C* and Ht are disjoint and so there exists a separating hyperplane Li. Since Zig belongs to both sets, it also lies in L”. For each i f N, we shall write L’ = {&+ qi * zi = q2 - z**>,
152
LEONID
HURWICZ
and assume (without loss of generality) that qi
. 9
2
qi
. z*i
for all
zi E Ci,
qi
. zi
<
@ . z*i
for all
zi E Hi.
Since Li is a supporting hyperplane for Ci, the monotonicity implies qi E R\+ for all i E N.
condition (c.3)
Suppose now that, for some r, s E iV, q+ # qs. Consider then the economy e’ = ((~9, ai)),,, where, for each i E N, the preference relation I? is generated by the utility function vi(zi)
=
qi
. zim
By (d’), the economy Z belongs to E. Also, because the indifference sets are hyperplanes parallel to Hi, z* is a Nash allocation for t?. However, since
i
22
x2<
I
Y Y2
01.
FIG. 4. Shading such as ‘tc_ is placed on the inside of the boundary of the set We’). Arrows attached to indifference curves show the direction of preference. A symbol such as R1 shows that the indifference curve so labeled corresponds to the preference relation IF.
ALLOCATIONS A'ITAINABLE THROUGH NASH EQUILIBRIA
153
individuals r and s have different marginal rates of substitution at z* which is in the interior of the Edgeworth Box, z* is not Pareto-optimal for 6 It follows that all the q” must be equal, i.e., ql
=
q2
=
. ..
= q” = q (say).
Consider now the economy e, = ((69, RLi)hEN where, for each i E N, the preference relation RL is generated by the utility function vyzq = q * 9. Again, by (d’) e, belongs to E and z* is a Nash allocation for e, (using the Nash Equilibrium message n-tuple m*). (See Fig. 4.) Hence, by (i), z* is individually rational for e, , i.e., q - (t.2 + z*i) > q * ci9
for all j E N.
(1)
Now the vector q defines at z* what Debreu calls “an equilibrium (for Z) relative to the price system q.” Suppose that equality holds for all j E N in (1) above. Then (q, z*) constitutes a Walrasian Equilibrium for Z, contrary to the initial hypothesis. Hence one of the inequalities in (1) must be strict, say q - (63 + z*i> > q * 69. (2) Summing
over j in (1) and taking (2) into account we get
4’ 1’cEN z*j >o which contradicts the feasibility requirement z*j = 0. Hence z* is infeasible and so not Pareto-optimal, contrary to the initial hypothesis. This contradiction completes, the proof of the Theorem. CjaN
COROLLARY TO THEOREM 2.
Replace condition (iv) by the following:
(iv’) for each i E N and each ml”’ E .A@(, hi(mi , m)(c) is a concave function of mi (with m)i(fixed). Then, with all other assumptions of Theorem 2 maintained, remains valid.
its assertion
Proof. It is sufbcient to show that (iv’) implies (iv). But this follows from the fact that iff: X -+ Y is a concave function on a convex subset of a vector space into a vector space Y, then the set F={yEY:y is convex.
154
LEONID HURWICZ
To see that this is so, consider x’, x” E X and
[al Y’ G .fW,
Y”
Also, let 7 = ey’ + (1 - e) y”,
o
Now, by concavity ofS, bl
ef(x’) + (1 - 6 f(x?
G f(x)
for x = 8x’ + (1 - 8) X”.
Also, by [al, [CI w + (1 - ew G efw
+ (1
0)f (x3.
Then fib] and [cc]yield ef + (1 - 0) Y” ~fm, i.e., J Gfc3. Hence F is convex.
2. ECONOMIES WITH PUBLIC GOODS Theorems 1 and 2, as well as the Corollary, stated above for pure exchange economies, have their counterparts for economies in which public and private goods are present and production is possible. In this section we indicate the modifications needed in the above proofs to obtain these counterpart theorems. In a slight change of notation, let x denote the vector of private goods and y the vector of public goods, both of one or more dimensions; both symbols represent, as before, net increments rather than total amounts. Where convenient, and without loss of generality, the first component of x will be used as numeraire. In the present notation, a preference relation R belonging to the class 93, is generated by a utility function of the form 4x9.Y)
=
wr,
+
+
e [ jz
Xl
+
(1
rj W-h,
-
eQ
+
do,,
xj
+
+
tj>
+
xi)
1
+
qs No,. .3
B * Y]
+
ys
+
kd]
ALLOCATIONS
ATTAINABLE
THROUGH
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EQUILIBRIA
155
where the index j runs over all private goods other than x1 and the index s runs over all public goods. In appropriate contexts, the x’s and related parameters have superscripts pertaining to individuals holding those goods; so do the parameters related to y’s, but the y’s-as public goods-have no such superscripts. As for production, it will be assumed to be characterized by constant returns. Let T denote the set of possible (x, y) production points. It is then assumed that T is a cone. I.e., ‘if (x’, y’) E T and h > 0, then (k’, Ay’) E T also. An element of Tcan also be written as (&., xi, y) since the x-component is aggregated from all individuals. The simplest familiar example, where x and y are each one-dimensional, is characterized by the production relation C&N xi + y = 0; in this case t i = -xi is the tax (contribution) paid by the i-th individual toward the production of y. The quantity-price combination ((x*~)~.~ , y*, p$, ( P$~)~~~-) is called a Linduhl Equilibrium for the economy e = ((wi, Ri)iaN , T) if (1) the quantity vector is feasible with respect to (w, T); (2) for each i E IV, Zig = (x*~, y) maximizes i’s satisfaction for preference Ri over the non-negative orthant of consumption subject to the budget equality p$ - xi + pzi . y = 0; (3) p$ * x*+p;*y* >P,* * x + Pz . y for all (x, y) E T, where we write x* = c xi and P,* = c pUi. ioN
iEN
(Since we shall be assuming strictly monotone preferences with respect to all goods, there is no loss in defining the budget constraint as equality. Also as mentioned above, the first private good will serve as numeraire, so that Pz, = 1.)
We may note that, because T has been assumed to be a cone, (3) in the preceding definition implies P,* . x*+p;*y*=o.
This is the familiar zero-profit condition under constant returns. Z(e) will denote the set of all Lindahl Equilibria for e; L(e) will denote the set of all Lindahl allocations for e. Bearing in mind the new interpretation of 92, , we can now state the counterpart, for public goods-constant returns economies of the type just described, of Theorem 1. THEOREM 3. Let E be a class of economies with public goods and constant returns technology T whose elements satisfy, mutatis mutandis, the conditions (a)-(d) of Theorem 1 and whose outcome function h satisfies conditions (i)-(iii) of Theorem 1. Then
-44
2 L(e)
for all
e E E.
156
LEONID
HURWICZ
ProoJ: The proof follows precisely the outlines of the proof of Theorem 1 and need not be repeated. It is sufficient to note that since, in that proof, we look at the situation of each agent separately, it makes no difference that the prices for public goods are “personalized.” It may also be noted that condition (c.1) of Theorem 1 could be weakened to permit zero initial endowments in the public goods. We now present the counterpart of Theorem 2. THEOREM 4. Let E be a class of economies with public goods and constant returns technology T whose elements satisfy, mutatis mutandis, the conditions (a), (b), (c), (d’), (e), (i), (iv) of Theorem 2. Then
B,, n &‘jj(e) C L(e)
for all
e E E.
Proof. Here again, the proof is parallel to that of Theorem 2. However, there is a divergence due to the different form of Pareto-optimality conditions when public goods are present. We shall therefore set out this phase of the argument in detail. We first proceed as in the proof of Theorem 2 to establish the existence, for each i E M, of a hyperplane Li separating Hi from Ci. Write pi = {(xi, y): ai . X’ + bi * y = ai ex*~
+ bi * y*}-
We shall normalize so that the first component of ai equals 1. Again, we consider the auxiliary economy S?where all preferences are generated by linear utility functions, with indifference sets which are hyperplanes parallel to L” for each i. As in the proof of Theorem 2, the initially given allocation z* is Nash for the economy E. It follows that a, = a, = -*- = a, = a (say),
since otherwise there would be unequal marginal rates of substitution at an interior allocation and this would contradict Pareto optimality of z* for the economy e’. Consider now the economy eL where, for each i E N, the preference relation RLi is generated by the utility function V'(Z)
Since z* is individually
= a * xi + bi *y,
rational, we have
a . x*( + bi - v* > 0
for all
i E N.
But if equality were to hold for all i E N in (l), z* would be a Lindahl
(1)
allo-
ALLOCATIONS
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157
cation. Hence the left-hand side in (1) must be positive for at least one individual. But then, summing over all individuals, we obtain a*X*+B.y*
>O
(2)
where X* = 1 x*i and B = 1 bi . ieN
iEN
However, we shall now show that (2) implies the non-optimality of z* for e, . This completes the proof since z* is Nash for e, . To show that (2) implies non-optimality, we construct an allocation that is Pareto-superior to z*. This allocation z** = ((x**i)iGN, y**) is defined as follows: y** = (1 + 6) y* where 6’0; x**j = x+-j forj = l,..., n - 1 where x#j is defined below; X**n
=
x**
_
12-l
n-1
cx**j=(l+e)x*j=l
i=l
Cx#j=(14).*--..+,#,
and x**
=
1
x**~
= (1 + 0) X* and X# = c J+.
EN
GN
In the preceding formulae, we define, for each i E N, the vectors x#{ by the equality (3)
It follows that z** is technologically feasible and that, except for individual n, everyone remains at the z* level of satisfaction. It is therefore sufficient to show that the n-th level of satisfaction is higher at z** than at z*, since this implies the non-optimality of z*. Now z**” is preferred by n to Zig if a*x**la-j-b;y**
>a*x#n$b;y**
which, after some cancellations, is seen to be equivalent to a . [(l +
e)X*
- P]
> 0.
(4)
On the other hand, summing (3) yields a . X# + B - ey* = a - x*.
(5)
158
LEONID HLJRWICZ
Solving for B * y* from (5) and substituting
into (2) then gives
a - x* + (l/@( x* - aX#) > 0.
(6)
(The division by 8 is legitimate since d is assumed positive.) Multiplying (6) by positive 0 does not change the direction of inequality. Hence,
which is the same as (4). The proof is complete. COROLLARY TO THEOREM 4. Finally, and without Q formal statement, we note that there is a Corollary ot Theorem 4 which is the exact counterpart of that for Theorem 2, with the identical proof.
3. ECONOMIES WITH PRODUCTION In this section we show that the results of Section 1 (where pure exchange was assumed and public goods were absent) can be extended to economies with production (but without public goods). A further extension to economies with production and public goods is straightforward, (Indeed, the model of Section 2 does involve production but under the special assumption of constant returns.) To describe the environment involving production, we write Tj to denote the production set of the j-th producer, j E J. An element tj of Ti, called a (net) output vector, has positive components for outputs, negative ones for inputs. An environment is then defined as e = ((wi, R&+ , (Tj)jsJ). An allocation s = ((z~)~~~, (tj)& is defined in terms of the net increments zi for consumers and output vectors ti for producers. Feasibility requires that the allocation be balanced, i.e., that
and, also, that each CJ + zi be in the individual consumption set (here assumed to be the non-negative orthant IJ!+~ of the commodity space). The n-tuple (&EN is called a distribution. An allocation s is, as usual, called Pareto-optimal if it is feasible and if there is no other feasible allocation whose distribution dominates that of s in terms of preferences. The set of allocations Pareto-optimal for e is denoted by ~(4. An I-tuple p* of reals is called an eficiency price (uector)fir the environment e at the allocation s* = ((z*~)~~~, (t*j)j,J) if s* is Pareto-optimal for e and if: (1) for each i E N, z*( maximizes Ri-satisfaction subject to the budget
ALLOCATIONS
AlTAINABLE
TliR0UGI-I
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constraint p* * 9 < p* * z*i, and (2) t*j maximizes (the efficiency profit) p* . tj for tj in Tj. [In Debreu’s terminology [2, pg. 931, s* is an equilibrium relative to the price system p*.] It is known that under certain assumptions of convexity, continuity, etc., every Pareto-optimal allocation s has an efficiency price associated with it; indeed, there may be more than one such price if the environment e is not “smooth.” We shall therefore find it convenient to introduce a function # selecting a particular efficiency price Jl(e, s) from the set (assumed non-empty) of the efficiency prices for e at s. The price (vector) #(e, S) will be called the selected eficiency price for e at s. It is perhaps not quite obvious what the appropriate generalization of the individual rationality concept should be. It is natural to seek a distribution (called the reference distribution) which would play here a role analogous to that played by the initial endowment n-tuple (ui)ieN in the case of pure exchange, as well as in the case of constant returns. (This approach is in line with the framework introduced in [Hurwicz, 4, pp. 18-191. A private communication by William Thomson concerning the analogues of Theorems 1 and 2 for “envy-free” (equitable) distributions was extremely helpful here.) Clearly, the reference distribution will depend on the environment. But in the case of production, it will also depend on how much was produced, by whom, and, perhaps other factors. We therefore introduce a function y, called the guarantee structure specifying the reference distribution given the environment e and the allocation s. If z* = (z*~)~~~ is the reference distribution generated by the guarantee structure y, we write z* = r(e, s), or z*( = yi(e, s), i E N. An allocation s# = ((z+QN , (t#&) is said to be individually rational with respect to the guarantee structure y for the environment e if and only if, for
where ki is the i-th preference relation (Ri) in e. Of particular interest is the$xed share guarantee strucfure where r(e, s) is given by f(e, s) = wi [ 1 + (P * ; e’(fj)/(p
. Ws] , Z-EN,
(1)
where p = #(e, s) is the selected efficiency price for e at s and the eji are nonnegative fractions such that xi 8{ = 1 for each j E J. We shall denote by $Je) the set of allocations that are individually rational with respect to the guarantee structure y for the environment e, and, as previously, we shall write S,(e) = g(e) n YJe). In particular, S,(e) and S,(e) will denote the corresponding sets with respect to the guarantee structure specified by (1). 642/21/1-11
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The game is again defined by the message (strategy) spaces JP and the outcome function h. However, here the outcome function must specify outputs as well as the distribution. We write, therefore, h = ((hri)ienr , (h&,) where hGi specifies zi and hf specifies ti, in both cases as functions of the n-tuple (m, ,..., m,), mi E .Mi for each i E N. As in the introductory section, we define, for each i E N and each rnji( E dPc ht(dP,
m)i() = (zi: zi = hzi(mi , rn)Q) for some mi E A?),
Hzi(m)“() = {zi: .zi < hzi(mi , m)i() for some mi E Ai}. Next, the set of Nash Equilibrium
strategies for the environment
e is defined as
vh(e) = {m*: for all i E N, hzi(m*) Rihsi(mi , m*ji() for all mi E Aft}, and the set of Nash allocations as A$(e) = {s: s = h(m) for some m E vh(e)}. We also need unambiguous notation for the familiar concepts related to the competitive equilibria of a private ownership economy in which the i-th consumer owns the share eji of the j-th production unit and is, consequently entitled to the corresponding fraction of its profits; here the i-th budget constraint has the form p’Zi
Although this formulation is due to Arrow and Debreu [l, pp. 270-2711, we shall give priority to considerations of uniformity in notation and terminology, and continue to refer to competitive equilibria and allocations as Walrasian. We shall denote by ?‘KO(e) the set of price-allocation pairs (p, S) constituting competitive equilibria for the environment e given the ownership structure specified by the matrix 19= (eji). The corresponding set of competitive allocations will be denoted by W,(e). Before stating, as Theorem l*, the counterpart for economies with production of Theorem 1, we need one more notational convention. In Theorem 1, we used the symbol EL,-(w) to denote the class of environments with the initial endowment n-tuple w and preferences from 9, or 9?L . In Theorem 1* and its proof, on the other hand, we shall use in corresponding places the symbol E&w, T) denoting a class of environments having the following property: Let e = ((wi, Ri)islv , (T&) be a given environment; let (tj)jEJ be a set of output vectors such that, for each j E J, the output vector tj belongs to Tj; let the vector pj define a hyperplane supporting Ti at tj; and, for each i E N, let RiEg’,Ug’,. Then the class of environments EL&w, T) contains an
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environment 8 = ((09, IP)i,, , (pj)j.J) such that, for eachj E J, tj is the unique point common to Ti and the hyperplane defined by pi. We may note that if Ti is “strictly convex at tj,” it then qualifies asr?‘i in the preceding definition. (We do not require that Tj be convex, but the requirement of the definition is vacuous unless Ti has a supporting hyperplane at tj.) Analogously, we define a class Et(w, T) of environments. Its definition differs from that of E&w, T) only in that the preferences all must come from B’r. , rather than from Br. u B’c. Let E be a class of economies e = ((CC/,Ri)i4N, (Tj>& with production satisfying the conditions of Theorem 1, with (d) replaced by (d*) in which E&w, T) takes the place of E&o). Furthermore let the outcome function h (as defined earlier in the present section) have the following properties: THEOREM
(i*)
1 *.
for each e E E,
(ii*) and (ii*), where the latter two conditions are obtainedfrom (ii) and (iii) of Theorem 1 by having ELc(u, T) and E,(w, T) take the respective places of E&w) and EL(w). Then
Proof. Since the proof is quite similar to that of Theorem 1, it will only be sketched here. Let e E E and d E W,(Z)). We shall show that SE Jr/-,(Z). First, we choose - _ price vector p so that ( p, s) E Y&(Z) and p = #(a, S); i.e., j is the selected efficiency price for Z for S. (Since dlrg(?) is non-empty by hypothesis, the price p is well-defined.) Now construct a sequence of economies e, from E&6, T), converging to some eL in EL(6, T), as follows. Choose preferences $” for e, just in the proof of Theorem 1 except that & must be replaced by the commodity bundle
where P is the j-th output component of S. As for technologies, let Fj be so chosen that pi defines a hyperplane supporting T’j at P, and ii is the unique point common to this hyperplane and T*. Then let, for each j E J and all v = 1, 2,..., Tvj = Td, and e, = @“, R%,N 9 (c%J).
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Given this construction it can be shown that the diameter of &(e,) + 0 as v --f co, and SE &(e,) for all v. By our assumptions, each e, has a Nash allocation S, and S, + S. Hence the closed graph property implies that s E M,,(e,) where eL = lim, e, and eL E EJw, T). But then, by reasoning analogous to that used in the proof of Theorem 1, it follows that s E .AqZ). Before stating, as Theorem 2*, the counterpart for economies with production of Theorem 2, we introduce a class of environments denoted by E,*(w, 7). Given an environment e = ((wi, R3rEN , (T&), this class consists of all the environments of the form d = ((ai, ai), (Tj)j,J) where Ri E 9ZL . I.e., it contains all environments with the same initial endowments and technologies as e while preferences are generated by linear utility functions. THEOREM 2 *. Let E be a class of economies with production s’atisfying the conditions of Theorem 2, with (d’) replaced by (d’*), using Ez(w, T) in place of EL(w), and the additional condition: (f) for all j E J, the set Tj is convex. Furthermore, let the outcome function h satisfy: (i*) for each e E E, B,, n &te) C X44; (iv*) f or each i E N and each rnji( E .A%‘)~(,the set Hzi(mji() is convex. Then, for every e E E, B,, n 4$(e) C W,(e).
Proof. Here again the proof follows the pattern of that for Theorem I, again with the right-hand side of (1) replacing wi as the “guaranteed” commodity bundle. It should only be noted that the common value q of the qi (“slopes of the separating planes Li) in that proof constitutes the selected efficiency price for the given Pareto-optimal allocation s* because eL has smooth (in fact, linear) preferences and so there is only one efficiency price at s* which must coincide with q. Thus let s* = (z*, t*) belong to X*(Z) and to S,(?) but not to W,(Z). We show that this yields a contradiction. For each i E N, let Li = {zi: qi . zi = qi . Zig} be the planes separating the sets Hi(m*)i() from the set Ci. Let e’ be the economy with the same initial endowments and technologies as 2 but with the i-th preference relation generated by vi@) = qi * z”. Again s* is a Nash allocation for ?, and so must be Pareto-optimal. This implies that all the qt must have a common value, say q. In turn, we consider the economy eL with initial endowments and technologies same as in Z but preferences generated by vi(zi) = q . zi. Again, s* is Pareto-optimal for eL and q is its (here unique) efficiency price. The requirement of individual rationality with respect to the fixed share guarantee structure 0 implies that
vyz**) > vi [Wi(q - T B,“t*j)/(q *cc+)]
for each i E N,
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i.e., that q . i!*i 2 q * 09 (q * c Bj’t*j)/(q
. C&)
for each i E N.
j
(2)
But equality cannot hold in (2) for all i f N for then (q, s*) would constitute a Walrasian Equilibrium for 2, contrary to the initial supposition. (It is obvious the budget equalities would be satisfied in virtue of (2) and that consumer satisfactions would be maximized because of the separating plane property of q. But profit would also be maximized by the efficiency price property of 4.1 Hence for at least one i’ E N, the inequality in (2) must be strict. It follows that when both sides of (2) are summed over i, the left-hand sum will be greater than the right-hand sum. Thus we obtain (after cancelling the term q in the right-hand side of (2) and summing over i)
q’ci
z*i > q . c c e,w z j
Now -& xj e,it*i = Cj t*j xi Bji = Cj t*j. Therefore (3) becomes
4.F
z*i > q . c t*j, i
(4)
which contradicts the feasibility (balance) condition x:f z** = Cj t*j. Since the latter must be satisfied by the Pareto-optimal allocation s*, a contradiction has been arrived at and the proof is complete. Remarks 1. We may note that, since “efficiency profit” p * tj = 0 under constant returns, the case where constant returns prevail retains the initial endowment as the “guaranteed” commodity bundle under the fixed share structure. Therefore, as in Section 2, production with constant returns does not require the more complex machinery of “guarantee structures.” 2. It appears that guarantee structures other than fixed shares could be handled by analogous methods. However, they would require appropriately modified formulae for sharing profits in a private ownership economy.
4. AN EXTENSION
TO NON-PLAYER
PARTICIPANTS
(QUASI-GAMES)
From the point of view of economic interpretation the preceding game structure is somewhat inadequate. In the presence of production it is natural to introduce information-processing (decision-making) entities (agents) who obey certain behavior rules but do not have preferences, and hence are not players in a game (“soulless” production managers). Even in the pure exchange case economists often find it convenient to introduce an auctioneer
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who is not a player. Of course, one may wish to endow the auctioneers or the production managers with preferences so as to take better account of incentive issues, but it is desirable to have room for non-player participants. This can be accomplished by defining what we shall call a quasi-game G. In this quasi-game there is a set of players N each of whom controls a strategy variable mi and a set of non-players K each of whom controls a decision variable dk in DL. The outcome function h associates an allocation as a function of the mi and the dk , say s = h((m& , (d,J&. Each player has a preference relation Ri on his component of the distribution in s. A non-player, by contrast, has only a behavior rule Fk which specifies the permissible values dk E DL given the strategies mi and the decisions d,f ED”‘, k’ E K - jk), and, possibly, also some information concerning the environment. (For instance, no such information is ordinarily assumed be used by an auctioneer, but it would be assumed known to a non-player manager of a production unit.) We express this by dk E Fk(m, dbk(, e)
k E K.
Now a Nash Equilibrium of the quasi-game G is defined as follows: given the strategies of other players and the decisions of the non-players, no player can secure a preferred outcome by modifying his own strategy; given the strategies of all the players and the decisions of other non-players, each non-player’s decision is permissible for the prevailing environment. Formally, (m*, d*) is a Nash Equilibrium of G for e if and only if; (1) for each i E N, hzi(m*, d*) Rihzi(mi , m*)i() for all mi E J@, and (2) for each k E K, d*” E Fk(m*, d*jk(, e).
A cursory examination indicates that our results can easily be extended to quasi-games of the type just described.
REFERENCES I K. J. ARROW AND G, DEBREU, Existence of an equilibrium for a competitive economy, Econometrica 22 (1954), 265-290. 2. G. DEBREU, “Theory of Value,” Wiley, New York, 1959. 3. T. GROVES AND J. LEDYARD, Optimal allocation of public goods: A solution to the “free rider” problem, Econometrica 45 (1977) 783-809. 4. L. HURWICZ, The design of mechanisms for resource allocation, Amer. Econ. Reo. 63 (1973),
l-30.
5. L. HURWICZ, Outcome functions yielding Walrasian and Lindhal allocations at Nash equilibrium points, mimeographed handout for presentation at a Stanford Seminar, November 21, 1976. (An earlier version circulated for Econ 208, Berkeley.) Reo. Econ. Studies, in press. 6. L. HUR~ICZ, Balanced outcome functions yielding Walrasian and Lindhal allocations at Nash equilibrium points for two or more agents, Boulder, Cola., September 3, 1977
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(private circulation). Presented at the Econometric Society meetings, Boulder, Colo., June 1978. 7. K. MOUNT AND S. REITER, The informational size of message spaces, J. &on. Theory 8 (1974), 161-191. 8. S. REITER, Information, incentive, and performance in the (new)z welfare economics, Amer. Econ. Rev. 67 (1977), 226-234. 9. D. SCHME~LER, A remark on microeconomic models of an economy and on a gametheoretic interpretation of Walras equilibria, mimeographed handout, Minneapolis, Minn., March 1976.