Nir Dagan, Roberto Serrano, and Oscar Volij

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We study a non-cooperative bargaining model in a continuum economy, in which coalitions of finite size are randomly matched. A triple equivalence between the core, Walrasian allocations and the strategic equilibria outcomes is established. Our proof makes use of the theory of the core with finite coalitions, indicating that non-cooperative bargaining in large economies is related directly to the core.

Typically non-cooperative models of bargaining assume that utility functions are concave. On the other hand, core equivalence hold under much weaker conditions, and even when some of the commodities are indivisible. To bridge this gap we intoduce a concept of aggregate risk aversion, and provide sufficient condition for it to hold in perfectly competitive economies. Aggregate risk aversion allows for indivisible goods and other non-convexities, but assures that the economy as a whole cannot benefit from intriducing random outcomes.

We study a Gale-like matching model in a large exchange economy, in which trade takes place through non-cooperative bargaining in coalitions of finite size. Under essentially the same conditions of core equivalence, we show that the strategic equilibrium outcomes of our model coincide with the Walrasian allocations of the economy. Our method of proof makes use of the theory of the core. With respect to previous work, our positive implementation result applies to a substantially larger class of economies: the model relaxes differentiability and convexity of preferences, and also admits an arbitrary number of divisible and indivisible goods.

**Keywords:** Finite coalitions and Edgworthian theory of exchange;
marginal rates of substitution and Jevonsian theory of exchange; matching and
bargaining; core; Walrasian equilibrium.

**JEL:** D51; D41; C78.

*Economic Theory* 15:279-296 (2000)

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From the July 1998 version

The Walrasian or competitive equilibrium is the central solution concept in economics. However, from its definition it is not clear what trading procedures lead to Walrasian outcomes. In contrast to what many economists may think, the original account of the theory (Walras, (1874)) does not rely on the existence of the so-called Walrasian auctioneer. Nonetheless, the usual formal presentation of the model includes the auctioneer implicitly due to a lack of explanation for the formation of equilibrium prices.

Negishi (1989) distinguishes two major schools in the analysis of markets. One of them considers prices as part of the economic mechanism in and out of equilibrium. This school is attributed to Cournot (1838) and Walras (1874). A second school, associated with Jevons (1879) and Edgeworth (1881), attempts to consider decentralized trading mechanisms and answer the question of whether equilibrium prices will emerge as the consequence of agents' trading actions.

We can distinguish at least two major approaches in the Jevons-Edgeworth school of decentralized trading. One of them, which today is referred to as the core equivalence literature, has its origins in Edgeworth (1881). This approach finds conditions under which core and Walrasian allocations are equivalent. If one conceives the core as a decentralized mechanism, these results give insights about the Walrasian allocations that were not provided by Walras. The shortcoming of this approach is that, although the core captures a natural idea of coalitional stability, it does not specify the trading procedure either. Here is relevant the work on the non-cooperative implementation of the core in finite games and economies (see, for example, Perry and Reny (1994), Serrano (1995) and Serrano and Vohra (1997)).

A second approach models the trading procedure more explicitly and studies its strategic equilibria. As part of a recent literature that starts with Rubinstein (1982) and Rubinstein and Wolinsky (1985) [henceforth, RW], many researchers have turned to models where a decentralized trading procedure is made explicit in a bargaining extensive form. RW (1985) analyze an assignment market and claim that in a frictionless economy the strategic equilibria need not be Walrasian. This claim was challenged later by the classic paper of Gale (1986a), who constructed an alternative bargaining procedure in a continuum economy in which strategic and Walrasian equilibria coincide. Gale's work was generalized in some respects by McLennan and Sonnenschein (1991) [McLS in the sequel].

Combining the above strands of the literature, in this paper we study a decentralized matching model in a large exchange economy, in which trade takes place through non-cooperative bargaining in coalitions of finite size. Under essentially the conditions of core equivalence, we show that the strategic equilibrium outcomes of our model coincide with the Walrasian allocations of the economy. Our model relaxes differentiability and convexity of preferences, and also admits indivisible goods. By considering multilateral meetings in which trade takes place, we are able to use the full power of the theory of the core. In doing so, it becomes apparent that the driving force behind core equivalence and Gale-like results is similar: core equivalence takes now a non-cooperative dimension.

From an implementation theoretic point of view, we construct a mechanism that fully implements the Walrasian allocations of an economy with a continuum of agents. In contrast to all previous work, though, implementation is achieved over a substantially larger class of economies.

Free of the differentiability assumption, our model differs from the preceding ones in allowing for trade to take place in coalitions with any finite number of participants, as opposed to only pairwise meetings. This trade-off in the modelling choice leads to two distinct methods of proof and captures different insights. The proofs of the existing results (Gale (1986a, c), McLS (1991)) rely crucially on the existence of the marginal rate of substitution at every bundle in order to get the unique (to that bundle) supporting price. Since in the strategic equilibrium of these models there cannot be any pair of agents with positive gains from trade, by differentiability, at the equilibrium bundles every two agents have the same marginal rate of substitution. Consequently, this must hold for all agents. Feasibility of the equilibrium outcome (zero aggregate excess demand) and some extra technical assumptions then take care of the rest of the argument. In contrast to this method of proof, Theorem 1 uses the theory of the core to show that every strategic equilibrium yields a Walrasian allocation. The existence of marginal rates of substitution is not necessary. Since we allow for finite coalitions to meet, in our strategic equilibria there cannot be any such coalition with positive gains from trade. Then, a powerful lemma due to Hammond, Kaneko and Wooders (1989) and core equivalence take us the rest of the way without the intermediate aid of differentiability.

Unlike our first result just referred to, its converse,
Theorem 2, does not use the theory of the core.
That is, although
we allow for finite coalitions, it turns out that all equilibrium
outcomes of our game can be supported by strategies according to which
all trade takes place in pairs. In this sense, pairwise trade arises as
an endogenous feature of the procedure instead of being imposed as an
exogenous restriction. On the other hand, our trading procedure
is indeed decentralized, since finite coalitions are negligible in the
continuum. Moreover, if all gains from trade in the economy can be
exhausted in coalitions bounded by
a given finite size *k* (e.g. assignment markets, like in
RW (1985) in which *k=2*), one can restrict
the matching
process to meetings of size *k+1*.

...

From the July 1998 version

By using the insights of the theory of the core, this paper presents a model of decentralized trade through bargaining in coalitions of finite size. This allows us to obtain equivalence results among core, Walrasian and strategic equilibrium allocations for a wide class of large exchange economies, including non-differentiable non-convex preferences and indivisible goods.

Our results are robust to several extensions of the model. First,
we could allow for any entry process (not necessarily one time
entry) as long as the measure of the total entering population is
finite. Second, different bargaining procedures in the coalitional
meetings could be adopted: for example, veto power can be given
only to those responders who are offered a non-zero trade vector.
Third, we only need to assume that an agent's probability of
meeting a coalition of size *n* be positive. In particular, we
could assume that the probability of meeting a coalition of more
than two agents be arbitrarily small and all our results would go
through. Thus, Gale's model can be viewed as the "limit" of ours
as the probabilities of multilateral meetings vanish. This poses
the important open question of lower hemicontinuity of the
equilibrium payoff correspondence, i.e., which of the extra
assumptions made by Gale are really needed to obtain the result
using only trade in pairs.

A separate dimension along which our results are more robust than the previous ones found in the literature is the class of economies to which they apply. We discuss this in length in the following paragraphs, especially comparing our results to the important works of Gale (1986a, b, c) and McLS (1991).

**Existence of equilibrium** Thanks to considering finite
coalitions in the procedure, our paper yields the equivalence
between strategic and Walrasian equilibria under essentially
the same assumptions as those needed for the core equivalence theorem.
Moreover, our assumptions also guarantee the existence of a Walrasian
equilibrium, as opposed to Gale (1986a,
b, c)
and McLS (1991), which
deal with open consumption sets.

**Limited applicability of the previous models** As
discussed in the introduction, we regard the relaxation of
differentiability as a crucial conceptual departure from Gale's
and McLS's work. From an applied view-point, the differentiability
assumption by itself that Gale (1986a,
c) and McLS (1991) make
is not very restrictive: many models in economics incorporate it
in order to allow for a closed solution and for the performance of
comparative statics exercises. However, the proofs of the above
mentioned authors rely on additional strong assumptions, that exclude
most applied models. Gale (1986a) assumes that
for each utility
function the support of the endowments compatible with it
is the entire consumption set. This assumption excludes the
possibility of a finite type economy. Gale
(1986c), who assumes a finite
number of types, uses a bounded curvature assumption,
thereby excluding, for example, Cobb-Douglas utility functions on the
non-negative orthant or its interior. McLS (1991)
make either a bounded curvature assumption similar to
Gale's (1986a) or a restriction on the
equilibrium which seems to require an assumption similar to
Gale's (1986a) on the primitives of the economy.
In contrast, our model, which applies to very general economies, also
applies to these standard cases.

**Feasibility in and out of equilibrium** We assume, like
Gale (1986a, b,
c), that the flow of agents entering the market
constitutes an economy, i.e., they sum up to a finite measure.
In addition, we also assume that short sales are not allowed. These two
assumptions together ensure that the flow of agents out of the
market is consistent with the feasibility constraint of the economy.
Suppose, like McLS (1991), that the total measure of
agents is finite,
but short sales are allowed. In this case, nothing assures that feasibility is
met. Consider an arbitrary assignment of bundles to agents, and
the following strategies (that do not constitute an equilibrium
in McLS's game with short sales). Each proposer asks for the bundle assigned
arbitrarily to him and each responder accepts any proposal; agents leave
the market as soon as they reach their assigned bundle. Clearly, these
strategies guarantee that each agent will get with probability 1
the assigned bundle. The problem stays even if we restrict attention
to the equilibria of their game. Indeed, the strategic equilibrium
that McLS propose (pp. 1395-1396) to support a Walrasian equilibrium is
a strategic equilibrium for any prices. That is,
for an arbitrary price vector, their strategic equilibrium
gives the outcome that every agent maximizes over the corresponding
budget set, but the market clearing conditions may be violated. This casts
doubt on the validity of such a model as a foundation of Walrasian
outcomes, since the IOUs are eventually consumed by the
agents. As we perceive market clearing conditions as an essential part of
the Walrasian concept, we do not allow for short sales and adopt Gale's approach,
which ensures feasibility in and out of equilibrium. See
Dagan, Serrano and Volij (1998) for other related
criticisms of the McLS model.

In the case where the sum of the measures does not constitute an economy, it is not clear what are the feasibility constraints. McLS's Theorem 3 deals with the case when the inflow and the outflow of agents have long run averages. Following this approach, we can define the flow of agents out of the market to be 'feasible' if it is consistent with the long run average of the inflow of agents. However, it is not clear to us how one can construct a model in which this kind of constraint is met in and out of equilibrium. Therefore, interpreting the result of RW (1985) as consistent with Walrasian allocations (as done by McLS's Theorem 3) is not sound: the outcome of their strategic equilibrium is consistent with a notion of feasibility, but behavior different from this equilibrium may violate the same feasibility notion.

**Strict concavity and indivisible goods**
McLS (1991) note the restrictiveness of
Gale's (1986a, b,
c) assumption
of strictly concave utility functions in a continuum setting. One should
expect that the convexifying effects of large numbers could be helpful
to relax this assumption. However, we believe that McLS's solution to the
problem is inadequate and provide an alternative treatment of the issue.
One difference between the underlying economy and the strategic model is
that in the latter the outcome (at least for an individual agent) may be
random and thus preferences on random outcomes must be specified. McLS
do not make any assumption regarding the concavity of utility functions;
instead, they allow for short sales, which enables them to prove
that outcomes of the strategic equilibria are not random. As explained
above, assuming short sales in this framework may be problematic.
In addition, a separate shortcoming of their treatment of
non-convexities is that they maintain the differentiability
assumption, which precludes non-convexities arising
from the existence of indivisible commodities.

Gale (1986a) uses the strict concavity assumption only to ensure that the introduction of lotteries does not enlarge the set of possible utilities of the agents. Thus, what is needed is a property of risk aversion in the aggregate. We impose a condition on the quasiconcave covers of the utility functions that ensures the sufficient degree of aggregate risk aversion. This assumption is compatible with having indivisible commodities as well as other kinds of non-convexities. Thus, our assumptions allow for a unified treatment of assignment markets à la RW (1985) and classical exchange economies à la Gale (1986a, b, c). We should stress that our assumption of aggregate risk aversion is sufficient to obtain Gale's results as well (of course, within his restrictive subdomain).

...

From the July 1998 version, with minor fixes.

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