An equivalence theorem for a bargaining set

An equivalence theorem for a bargaining set

athematical Economics 18 ( 1989) 129-139. North- ndreu Harvard University, Cambridge, MA 02138, USA Submitted May 1987, accepted June 1988 First, a ...

1MB Sizes 1 Downloads 11 Views

athematical Economics 18 ( 1989) 129-139. North-

ndreu Harvard University, Cambridge, MA 02138, USA

Submitted May 1987, accepted June 1988 First, a modification of the Aumann-Maschler Bargaining Set is proposed. Then it is shown, under conditions of generality similar to the Core Equivalence Theorem, that the Bargaining Set and the set of Walrasian allocations coincide.

A weakness of the Core as a solution concept in economics and game theory is that it depends on the notion that when a coalition objects to a proposed allocation, i.e., engages on an improving move, it neglects to ta into account the repercussions triggered by the move. See Greenberg (1986) for a recent discussion of s point and its connection to the ideas underlying the von Neumann orgenstem stable-set solution. aschler ( 1964) pro osed a solution concept, the ses the above iss . Their idea is that for a based on a coaliti e effective it must be justified by the absence of a ns vrrhich admit counterobjections are frivolous and, therefore, disregarded. Counterobjections are fined as proposals which are improving for some other coalition but whit however, guarantee that any common member of the two coalitions will be as well off as with the objection. This is, of course, an im recise description of the idea and, in fact, a unique definition of the Bargaining Set. From t concept several useful variants have ev en (1982) and Shubik (1983). t is larger than the Core: blocking is ha Shapley and Shubik [see Shubi *My first debt is to L. Shapley. It was *becauseof his talk at the April Workshop on the Equivalence Principle (organized by A. Neyman) th conference. Conversations and correspondence with 3. Gabszewicz, B. Grodal

itutions. Financial support fr earth supported in part by NS 0304-4058/89/$3.50 0 1989,Elsevier Science

ort

1130

A. Ma.+CoOell,An equivalence theoremfor a

on distinguished individual players and it is thus well defined in the continuum case. 0ur proof is not in essence more complex than Aumann’s [email protected] it is different and it includes an existence argument. The key idea is a characterization of justified objections as those objections that, in a precise ur proof puts together t central sense, can be price alysis, namely those that underline, ctively, arguments of equilibriu ndamental theorems and of the existence of the proofs of the two equilibrium. It is possible that the proposed redefmition of the argaining Set be of more general interest. To assess this, what is required is to see which forms the Bargaining Set takes in general games or in economic environments , situations where the uivalence principle does not ere the Core is empty. itz (198811 is that the equality of d not hold in cases where there allocations [as in Shitovitz (1973111. ning Set discriminates better titive environments. ly competitive and non-co on general games.

set of agents is I= [O, l]. space of continuous and y is a (measurable) map 11seconomy remains fixed $ such that Ixsjw. such that, for a.e.

Lebesgue measure strictly monotone t for e allocation

t d,:p*x(t)

~Spw(t)

x is and

n equivalenw

(b) y(t) k,x(tl efinition

where Tel

2.

theoret; for

0

Bargaining Set

131

for all a.e. t&S and A{t~S:y(t) >,x(t)}>O. Let (S, y) be a objection to the location x. The pair (r z), and z: T-, is a counterobjection to (S, y) if:

(a) ~&STo, (b) A(T)>O, and (c) (i) z(t) >&t) for a.e. t E TnS, (ii) z(t) >,x(t) for a.e. t E T\S. S, y)

is said to be justified if there is argaining Set is the set of allocations against which there is not justified objection. alrasian allocation necessarily belongs to the there can be no objectio justified or not, against it (just ap argument showing that alrasian allocations belong to the of this paper is to show that the converse is also true Theorem 1. Zf there is no justified objection against the allocation x then x is Walrasian. Hence, an allocation x belongs to the argaining Set if and only if it is Walrasian.

bserve that even the weaker result: ‘if there is no justified objection against JCthen x is ret0 optimal’ requires a non-trivial proof. A topic for fu research is th\= tijgimptotic version of the above theorem. Note inat our argaining Set ii well defined $-)r finite economies and that it can well be la r tbn the Care (see exampk sec:“,ioll6). rd in equilibrium Although the hypotheses of the Theorems ard st Core equivalence theory they are stronger than whst is theorem in two respects. First, we quire strict monotonicity of preferences is is because establishing existence of a and strictly positive endowments. type of equilibrium is t of our proof. Se remains valid extent to which transitive preferences. c-- ferences are unordered has been inv (1989) has studied the case where con convex. Related concepts of argaining Sets for economies Yind ( 1986).

132

A. Ma+CMi,

An equivalence theoremfor a BargaitiingSet

refu

simply be replaced by weak preference (as it w3 consideration of the example in resence of weak pre Theorem 1 will still obtain

we define a S-objection to be a tr S9y) is an objection with y(t) >,. , S, y) is a counterobjection (r z) to n is justified if there is no counterobjection to it. Let there be a justified objection

latter is justifie ‘given any 6 > 0, if there is no then x is alrasian’.

(S, y)

against x. or any 6>Q we can 4, and transfer some of their t is a d-objection (K,s, y’) which this implir;s that any counter:Bunterobjection to (S, y).Since the also. Therefore Theorem 1 yields: bjection against the allocation x

he central idea for the roof of Theorem 1 is a consideration of a special class of objections generated by means of prices. efinition 4.

is

lrasian if there is

a price system p (i) p9.Qp=o(t) (ii) j.wQp-o(t)

for v&y(t), for v&x(t),

t&Y tEZ\S.

have a self selection is formed by precisely vector p than get the

hsian

objection (S9y) to an a&location x is just$ed. annotation then there is a

basian

A. Ma&dell,

An equivalence theoremfor a Bargaining Set

133

osition,the converse of 1, shows that the alrasian objection is more than a technical toal. Proposition 3. If (S, y) is Qjusttfied objection to an allocation x then it is also a badan objection.

o= x one gets, as a Corollary of sian allocation for the entire economy. Remark 4.

osition 2, the

Propasitic

ar that typically there will an objectionable allocati Indeed justified rasian and, while those exist d demand problem ote the contrast with well known that under the conditions of rnaosition 2 if allocation is not Walrasian then it can be improved upon in grda! variety of manners. t

Remark 5.

he fact that justi bjections must be alrasian also helps to understand why we cannot stre then the concept of objection by requiring strict preference for a.e. t E S. pose we are in a type economy and we consider an ahocation x satisfying the equal treatment property. Then the tion (S, y), with objecting price vector p, will itself satisfy the property in the following sense: if t, t’ are of the same type then y(t) mt x(t). This is easy to verify. Vote first that alrasian allocation for S (which implies p y(t) = p o(t) because y must be also p ~0). Suppose now that y(t p(t). Then y(t) >,#x(t’) and so p-y(t)> p o( t’) = p o(t) which is imposs ence y(t) - t x(t)* e must conclude therefore that if a coalition with a justified objection includes only part of some type of agents (and this may be unavoidable; see next two remarks) then it is not possible for these agents to strictly improve at the objection. l

l

l

l

Remark 6.

Given an allocation x and a price vector p any agent t having a to x(t) will be rasian demand at p strictly preferred (resp. disprefe asian objection (resp. remain outside) of any group attempting a freedom left corresponds to the ma sustainable by p. ent between getting x(t) or t consumers i who demands at p. may be instructive to d

134

A. Madolell,

An equivalence theoremfor a Bargaining Set

of 2

alrasiard objecuon can only have one of the

0 (ii)

A.

Mas-Cole&An

equivalence

theoremfor a BargainingSet

135

could be improved upon (even strictly) by a coalition using an allocation alrasian for the coalition had been proved before [in roposition 7.3.2, as an extension of a result of Town replica case]. The ‘stence of a Walrasian objection is, how stronger property. example, going back to fig I it should any price intermediate p2 can be used to generate a coalition improving upon at alrasian allocations. If, as a matter of definkion, we restricted counterobjecting groups to be subsets of the corresponding objecting groups then Theorem 1 remains valid (objecting is now easier!) and justified objections would be characte~~d by bein Walrasian allocations for the objecting coalition.

oofof0 Proof of Froposteion 1. The proof is just a repetition of the familiar argument establishing the optimality of equilibrium allocations (i.e., the *first fundamental theorem). Let p be the price vector associated with the lrasian objection (S, y). ause is ysLo and p-y(t) 2p-a(t) for a.e, te S we have that y is a

lrasian allocation for S with price vector p. This also implies that p>>C [remember that preferences are strictly monotone and o(t)>>O]. Suppose there is a counterobjection (?: z) to (S,y). Then for a.e. t E Tn S we have z(t) &y(t) and therefore p-z(t)&y(t)zp-o(t), with strict inequality if z(t) >,y(t). For a.e. tE T\S we have z(t) &x(t) and therefore p’z(t)2p*w(t), with strict inequaliy if z(t) >,x(t). e conclude that ~Tp*z(t)>~Tp-~t) which contradicts Irz <=j’rw. [II Proof of Proposition 2. The proof amounts to an adaptation of the familiar arguments establishing the existence of equilibrium in economies with a continuum of traders and non-convex preferences. It is analogous to the (1985). proof of Proposition 7.3.2 in p #O, denote C(p) = (t: there is u such Let x be a allocation. For &x(t) and p - u cp - co(t)}. Observe that if A(C(p))=0 for some p then x th lrasian. Hence we assume from now on that il(C(p)) >O for all p. is note B= {pi RI: llpll= 1,p>>O} and let f: correspondence generated by our economy. demand correspondence S*: f*(P,

0 = S(P% 0

136

A. Mas-Colell, An equivalence theoremfor

a Bargaining Set

Finally, put F* (p) :=s f*(p, t) dt. The correspondence F following standard properties: (i) p F*(p) = 0 for hemicontinuous and bounded below, (iii) F every p, and (iv) if pm+p, #=O for some j, and v, this is easily verified; see as-Cole11 (1985, p. 2 nical facts. ‘Note that the proof of (iii) requires heorem) even if preferences are convex. Lyapun monotonicity and &C(p)) > 0. follows The aggregate excess demand correspondence F* satisfies, therefore, all the conditions for the existence of an equilibrium, i.e., there is p>>O such that F*(p) =0 [see, e.g., ebreu (1970) for the argument]. Let w: 142’ be such E f(p, t)} and define sw=O. Take S=(t: that w(t)E:f*(p, t) for all teZ e claim that (S, y) is a ahsidn objection to y: s-, : by y(t)=w(t)+o(t). x with the price vector p. Indeed: (i) 2(S)> 0 because C(p) c S, (ii) sSy Sls CQ because j o =0 and, also, w(t) =0 whenever t $ S, (iii) if t E S and v &y(t) then v&, f(p,t)+o(t) and so p-v&p=o(t), (iv) if td\S then x(t)k,f(p,t)+w(t) 0 and so v k&t) implies p- v2p-o(t). l

Proof

of Proposition 3. It amounts to a variation of the familiar Schmeidler’s proof of the Aumann equivalence theorem [see, e.g., ildenbra :d (1974)] which can in turn be viewed as a sophisticated version of the second fundamental theorem. by w(t) = y(t) if t E S and w( x(t) if t $ S. Lef then Lyapunov”; theorem tw(t)}u{O) and V=s V(t)dt. as-Cole11(1985)] the set V is convex. If V n( -R’, +) # 4 then there would be a counterobjection to (S, y). Therefore we can assume that 0 $ j S. If we now let p 540, support V at 0 we have that, for t, p v 2 p- w(t) whenever v & w(t) which, of course, means that (S, y) is a rasian objection. 0 l

This section is in the spirit of an extended remark. The following definition iterates one more step the objection-counterobjection logic. Definition 5. Let (r z) be a counterobjection to the objection (S, y) to an allocation X. e say that (7’~) is justified if there is no (K v) such that J(V)>O, &,v~S~~o and, for te F:

v(t) >,z(t)

if

v(t) >ty(t)

if ZE VnS,

v(t) >,z(t)

if 245

ZE

A. Mm-Co6el1,An equivalence theoremfor a Bargaining Set

137

Suppose now that we modify our deli ition of counterobjection in section to allow for weak preference (except for a positive measure subsets of T). en we can strengthen our results by showing that any objection which its a counterobjection (i.e., any nonlrasian objection) admi fact a justified counterobjection. The key is following concept of rasian counterobjection: 2

Definition 6.

allocation x is

The counterobjection (‘I:z) to the objection (S, y) to the alrasian if there is a price p # 0 such that, for a.e. t,:

(i) pv>=pm(t) (ii) p-vzp-w(t) (iii) p*vzp-w(t) The proofs of that, respectively:

for for for

v&z(t), v&y(t), v&x(t),

tET tfd, td\(SuT).

s 1 and 2 can then be easily adapted to show is justified, and (ii) if an objection is not Walrasian (hence, by roposition i, not justified) then there is a Walrasian counterobjection against In the proofs of the two Propositions we only have to replace x by a x’: Z-R: defined by x’(t) = y(t) if t E S and x’(t) =x(t) t $ S. Substitute also the term objection by counterobjection and the symbols S, y, ‘I:z by, respectively, K z, V;v. In a recent interesting contribution Dutta e* al. (1987) have pushed a logic similar to this section to its limit and defincJ B Consistent There is little doubt that this is conceptually right and therefore we want to emphasize that none of these refinements and extensions detract from the strength of the Equivalence Theorem. 6n the contrary, by making counterobjecting easier we only make it more difficult for the equivalence to hold! rasian counterobjection

is well defined for general games. Suppose ‘[email protected] the characteristic form of a et of payoffs that coalition S can attain by itself. Suppose that XE V(N). e say that (S, y) is a 2nd y, 2x, for all t E S, where at least one of the inequ countcrobjection to (S, y) is then a (r z) such th z, 2 x,) for t E T n S (resp. t E T\S). At least one of strict. An objection is justified if it has no counter imputation if it is weakly optimal (i.e., x’>>x implies Set if formed by the set of i objection (important note: we vidaiy rational).

138

A. Ma-Cole& An equivalence theoremfor a Bargaining Set

xcept for the addition of weak optima ove is the exact analog of now very little about th not difficult to verify th rnel has the same definition than the (1983) - except that individual rationality is prenucleous and it is always non-empty) and superadditive general games the set may be empty tive case. In fact counterexample to existence is yet available for the super it appears that in many examples the re can be quite large. Setting these issues se has recently been ma by Vohra (198 following the cue of aschler et al. (5972), it is shown tltlat for ordinally ning Set is no larger than the Core. a transferable utility example showing that for totally with at least four persons) our argaining Set may indeed be larger than the Core. This is of special interest to us because totally balanced games car be generated from exchange econo,zGesfitting the framework of section 2 [see Shapley and Shubik (1969)]. Example. I = {1,2,3,4} The game v:2’4 R is normalized to v(t) = 0 for all t ~1 and is the minimal superadditive game compatible with the values v(1234)= v(123)=3.1 and ) = ~(34)32.06. The game is totally balanced (a Core putation is (0,l 9,0.2)). The imputation (1, 1, 1,l) does not to the Core since it can be improved upon several coalitions. It o be seen that it does not belong to the gaining Set used by Shapley and Shubik [see Shubik (1984)]. Nonetheless, in our sense every objection against it admits a counterobjection and it thus belongs to our relation of our definition of the the one used by Sha the relation to the arability. This is because we make counterobjecting easier (no member of the objection coalition is ng to a counterobjectin group) but on the er by requiring that t least one of the ning the counterobjection be strict.

A Mas-Cole& An equiualencetheoremfor a

rgainingSet

139

onomies with a finite set of equilibria, Econometrica 38, 387-392. wSengupta and R. Vohra, 1987, A consistent bargaining set, Journal of Economic Theory, forthcoming. Geanakoplos, J., 1978, The bargaining set and nonstandard analysis, ch. 3 of Ph.D. dissertation (Harvard University, Cambridge, MA). Greenberg, J., 1986, Stable standards of behavior: A unifying approach to solution concepts, Mimeo. (Stanford University, Stanford, CA). Grodal, B., 1986, Bargaining sets and Walrasian allocations for atomless economies with incomplete preferences (MSRI, Berkeley, CA). Wildenbrand, W., 1974, Core and equilibria of a large economy (Princeton University Press, Princeton, NJ). Maschler, M.. 1976, An advantage of the bargaining set over the core, Journal of Economic Theory 13, 184-194. Maschler, M., B. Peleg and L. Shapley, 1972, The kernel and bargaining sets for convex games, International Journal of Game Theory 1,73-93. Mas-Colell, A., 1985, The theory of general economic equilibrium: A differentiable approach (Cambridge University Press, Cambridge). Owen, G., 1982,Game theory (Academic Press, New York). Peleg, B., 1986,Private communication. Shapley, L. and M. Shubik, 1969,On market games, Journal of Economic Theory 1,9-25. Shapley, L. and M. Shubik, 1984, Convergence of the bargaining set for differentiable market games, Appendix B in Shubik, 1984,683-692. Shitovitz, B., 1973, Oligopoly in markets with a continuum of traders, Econometrica 41, 467-501. Shitovitz, B., 1988, The bargaining set and the core in mixed markets with atoms and an atomless sector, Journal of Mathematical Economics, forthcoming. Shubik, M., 1983,Game theory in the social sciences (MIT Press, Cambridge, MA). Shubik, M., 1984, A game theoretic approach to political econbmy (MIT Press, Cambridge, MA). Townsend, R., 1983, Theories of intermediated structures, in: K. Conference Series on Public Policy, Vol. 18, 221-272. Vind, K., 1986, Two characterizations of bargaining sets, manuscript (MSRI, Berkeley, CA). Vohra, R., 1989, An existence theorem for a bargaining set (Brown University, Providence, RI). Yamazaki, A., 1989, Equilibria and bargaining sets without convexity assumptions, Discussion paper (Hitotsubashi University, Tokyo).