"In many real markets, the competition is far from perfect; such markets are probably best represented by a mixed model, in which some of the traders are points in a continuum, and others are individually significant...[t] he chief result [core equivalence] holds only for a continuum of traders." (Aumann, 1964: p.41)

Or does it? This is where the Edgeworthian "numbers" theory of competition begins to get sticky. It is perfectly conceivable that we have a handful of "big players" (e.g. A) cohabiting with an "infinity" of small ones (H\A). This is a rather interesting situation. Intuitively, the original Cournot (1838) appeal to large numbers was to "drown" the effects of big players, i.e. to ensure that nobody could influence their own price. Edgeworth (1881) borrowed that theory for himself and applied it to the coalition-making process of the core. But there is an interesting differene of emphasis between Cournot and Edgeworth. If asked as to what is the "true" meaning of perfect competition, we would expect Cournot to reply "no atoms" and Edgeworth to reply "an infinite number of agents". For Cournot, the infinite numbers of agents was merely a device to guarantee perfect competition; for Edgeworth, infinite numbers was the definition of perfect competition. Surely, Cournot would not consider a continuum economy with atoms -- this oligopolistic "mixed-market" -- as perfectly competitive, whereas Edgeworth might.

However, there has been doubt as to whether an economy with some "atoms" is really the appropriate representation of Cournotian oligopoly. An "atom", after all, is an enormous person, not a firm. Furthermore, it is a person that is so "big" -- with infinitely greater endowment than anyone else in the economy -- that we can hardly call him a "person" at all. Werner Hildenbrand expresses his doubts:

"No doubt a rigorous interpretation of a model with "big" and "small" traders is possible. However, in a model with pure exchange, where there is no production, where all goods have to be consumed, and where there is no accumulation, it is difficult to see the economic content of such an interpretation." (Hildenbrand, 1974: p.127)

Aware of this difficulty, Jean Gabszewicz and Jacques Drèze (1971) have suggested the alternative interpretation of an atom as a "syndicate" of traders, i.e. a coalition of atomless agents whose members delegate all decision-making power to a representative. Thus, an "atom" acts like a single person, even though it really is an (indivisible) group of agents. However, as Hildenbrand (1974: p.126) points out, such an interpretation is not quite adequate for we would have to assume that every member of this syndicate would have the same preferences and endowments and that each would receive the same allocation. This is not self-evident in the definition of a syndicate.

While tempted by the "syndicate" interpretation, we prefer considering the atom to be a "big" person. While heeding Hildenbrand's caution that an "atom" in a pure exchange Edgeworthian economy may not look very much like a Cournotian firm, we must nonetheless realize that it is a "thing" (OK, a person) with market power. And market power is what matters in defining "oligopolistic" markets, no matter who or what yields it.

How can this be? Shitovitz's theorem explicitly assumed that the "atoms" in the economy were all of the same "type" but not the same size, i.e. they have the same preferences and same (scaled-for-size) endowments. As the atoms are of the "same type", then we can think of them as competing large oligopolistic firms with absolutely no difference in their wares (it's not even Coke vs. Pepsi, but more like a Coke vendor on your left and a Coke vendor on your right). The competition between them would be so intense that any gain they could make because of their size disappears. To take Shitovitz's example, a large hundred-room hotel in a town would be so threatened by a tiny three-room hotel (with rooms of the same quality) that the prices for all the rooms would be driven to their equilibrium price immediately. As long as another hotel exists, the big hotel would lose all of its market power!

This is not immediately obvious and it arises from a rather silly knot in the concept of the Edgeworthian recontracting. Why couldn't the large hotel charge, say $100 per room and let the small hotel charge $50, knowing that the latter would run out of capacity soon? Because in Edgeworthian exchange, the contracts are tailored to names. So, if the big hotel offers a hundred people (call them Mr. One to Mr. Hundred) each a $100 room while the small hotel offers three other people (call them Mr. Hundred-and-One to Mr. Hundred-and-Three) a room for $50, is this allocation blockable? Yes, because the small hotel can drop its own clients and offer Mr. One, Mr. Two and Mr. Three each a room for $70. As these people used to be clients of the big hotel under the initial allocation, they will accept the small hotel's cheaper rooms. Consequently, the initial allocation is "blocked."

Why does this matter to the big hotel? Why shouldn't the big hotel just replace its lost clientts with the clients that the small hotel dropped when forming its coalition? If it did so, it would have no incentive to drop its price. But this is where a rather ridiculous aspect of Edgeworthian exchange comes in: such a move is not allowed in the story! As long as any coalition can block an allocation, it will block, and that's that. The fact that Edgeworthian contracts are "tailored to name", as opposed to being open for all comers, prevents the large hotel from replacing the lost clients. All that matters is that the big hotel lost Mr. One, Mr. Two and Mr. Three, not that it could have, in compensation, gained Mr. One-Hundred-and-One, Mr. Hundred-and-Two and Mr. One-Hundred-and-Three. That is enough to block. There is no way, then, that the big hotel can maintain its clientele other than by offering every one of its clients a room at the competitive equilibrium rate to ensure that none of them leave it for the smaller hotel.

Another intuitive explanation for the Shitovitz theorem is given by Joseph Greenberg and Benyamin Shitovitz (1986). Imagine an economy where the atom A is not really an atom but actually a whole bunch of tiny individual traders who happen to have the same preferences and endowments (a variation on the Gabszewicz-Drèze "syndicate") If so, they might act "as if" they were an atom, loosely speaking. Now, Aumann's theorem places no restriction on the type of preferences or endowments agents are allowed to have. Thus, in an economy without atoms, but with a whole bunch of people who, together, "act like" an atom, Aumann's result should still be true.

It is necessary to be careful about what is meant by "acting like" an atom means, so we ought to be a little more precise. Let H be the atomic-mixed economy -- where A is the set of atoms (all with same preferences and scaled endowments) and H\A is the atomless part. Without loss of generality, assume that H\A = [0, 1]. Now, let us construct a second economy, call it H*, where we "split" the atoms into atomless syndicates. This is shown in Figure 2, where we take atoms a₁ Î A and a₂ Î A from economy H, split them into tiny agents with identical preferences and endowments, and so construct a new economy H* which is composed of the original atomless part (H\A) plus the atoms-turned-into-atomless-agents part. Intuitively, from an economy H with a big Robinson Crusoe, a big Friday and a bunch of atomless agents, we construct an economy H* with a bunch of tiny Robinsons, tiny Fridays and the rest of the old atomless agents.

Fig. 2 - Splitting the Atoms

Are these economies, H and H*, different? At first glance, yes -- and they ought to have different cores. To see why, recall that in the completely atomless economy H* you can make a coalition S* composed of, say, no mini-Robinsons, two-fifths of the mini-Fridays and two-thirds of the atomless others. Such a coalition seems to have no counterpart in the atomic mixed economy H because we cannot "split" the Friday atom to form it.

But there is a counterpart. By a result of Karl Vind (1972) on restricted cores in atomless economies, we know that if a coalition S* blocks allocation x* with allocation y*, then we can find a coalition of any size, say T*, to block x* with y*. So, suppose we "expand" the coalition S* into T* where T* has no Robinsons, all the mini-Fridays and three-fifths of the others. Now, here is the trick: when coalition S* was formed, two-fifths of Fridays got y*(h) and, by the rules of a blocking coalition, it was preferred to the original allocation, y*(h) >_h x*(h). But now that we have "expanded" our coalition to T*, what do the other three-fifths of the Fridays get under y*(h)? Presumably, they must get something that is not worse than the original allocation x*(h) in order join coalition T*; so let us say, for the sake of argument, that they get precisely x*(h). So, the average mini-Friday gets (2/5)y*(h) + (3/5)x*(h).

If we were in a finite economy and there were n mini-Fridays, then all the mini-Fridays together get n[(2/5)y*(h) + (3/5)x*(h)]. But this is not quite right as there is an infinite amount of mini-Fridays in H*. So, properly, noting that m (F) is the relative size of the original Friday atom before it was split, then all the mini-Fridays form m (F) of the total economy H*. So, the mini-Fridays altogether receive [(2/5)y*(h) + (3/5)x*(h)]/m (F).

Now, turn back to the atomic economy, H. Under the proposed allocation, the Friday atom would get x(F); but he is now considering an alternative allocation y(F). Let y(F) = [(2/5)y*(h) + (3/5)x*(h)]/m (F). Note that y(F) is precisely what all the mini-Fridays got together under coalition T* in the atomless economy, H*. So, how do we know the atom Friday will prefer y(F) >_h x(F)? Well, recall that x(F) = [(2/5)x*(h) + (3/5)x*(h)]/m (F), where x*(h) is the amount a single mini-Friday would receive originally. As Friday has convex preferences and we know that y*(h) >_h x*(h) for the lucky mini-Fridays (who all share the same preferences as the Friday atom), then y(F) >_h x(F). Thus, the Friday atom has found an allocation which he prefers to the original. Thus, in the atomic economy, the atom Friday, together with the two-thirds of the atomless agents who were present in both economies, will form a coalition T (corresponding to T*) to block x with y.

In sum, whatever allocation the coalition S* of mini-Fridays with the assistance of other atomless agents blocked in H*, the big Friday atom with these same others can block in H. How do we know we can always do this? What if the original coalition S* in H* included two-fifths of the mini-Fridays, two-thirds of the atomless agents and one-sixth of the mini-Robinsons? What now? That is easy. As we have assumed that all atoms are of the same type (same preferences and (scaled) endowments), then the one-sixth of the mini-Robinsons are identical to an equivalent amount of mini-Fridays. So, we just reconstitute the coalition so that it is all composed of "mini-Fridays" and atomless agents. Then we proceed as before: scale the coalition to T* so that the proportion that is made up of Fridays and Robsinsons-turned-Fridays is of the same size as the original Friday atom. We have reconstituted a blocking coalition which contains the Friday atom.

Perhaps some graphical intuition may help. Consider the simple two-type replica economy, reproduced in Figure 3. This is the type of economy Michael J. Farrell (1970) worked with. With two agents only (one Robinson, one Friday), the core extends from allocation B to A. In a n-agent replica economy (n Robinsons, n Fridays), we know that allocations A and B are blockable.

Fig. 3 - Blocking Coalitions

Now, allocation A, as we know, can be blocked by two Fridays and one Robinson -- each of the Fridays getting G and Robinson getting A. Increasing the size of the replica economy to n, does not affect the blockability of A: in an n-replica economy, any two Fridays and any one Robinson can continue to block A. Furthermore, for any m £ n/2, we can always find 2m Robinsons and m Fridays to block A -- just match two Robinsons to each Friday and make m such combinations. Thus, a coalition composed of n Fridays and n/2 Robinsons can block A.

Now let the all n Fridays constitute "one atom" -- so, we have n Robinsons and one gigantic Friday of size n. Can A continue to be blocked? The answer is, of course, yes. This is because the one big Friday can always take the n/2 tiny Robinsons to block A. So, the blockability of A is not in question in an atom economy -- as long as there are n/2 mini-Robinsons in it.

Ah, you may object, but what about allocation B? This is where the trick does not really work. In a replica economy, n Robinsons and n/2 Fridays can always block B, but now when we reconstitute the Fridays into one big atom, then B is no longer blockable. The Robinsons cannot exploit a split in Friday anymore; its as if all n Fridays formed an indivisible united front against them.

This is why we need at least two atoms of the same type (Farrell, Shitovitz) and/or some group of atomless people of similar type (Gabszewicz-Mertens). In Farrell's (1970) original contribution in a replica economy, he fixed the number of Fridays at n ³ 2 and let the number of Robinsons increase to infinity.

So, suppose we have an economy with n mini-Robinsons and two n/2-sized Friday atoms. In this case, the n Robinsons could combine with one n/2-sized Friday to block allocation B. Notice that the relative size of the atoms wouldn't matter; they'd just play one Friday atom against the other in a manner akin to the Shitovitz two-hotel case. So, if we have two Friday atoms, one of size n/4 and the other of size n3/4, all we would have to do to block B would be to take, say, n/2 Robinsons and combine them with the n/4-sized Friday atom. In Gabszewicz-Mertens, alternatively, even if we only had one big Friday, the Robinsons could still block B by combining with the other atomless people of the same type (a bunch of mini-Fridays). This would be a situation akin to there being one big hotel and lots of tiny little motels.

The cleanest proof of this theorem, due to Joseph Greenberg and Binyamin Shitovitz (1986), employs precisely this logic of "transforming" the atomic economy into its atomless counterpart by splitting the atoms. We follow it here.

Theorem: (Oligopolistic Core Equivalence) Let E: (H, Á , m ) ® Ã ´ Rⁿ₊ be an economy with a continuum of agents. Let A denote a set of "atoms", where if a Î A, then m (a) > 0. We assume #A ³ 2. H\A is the set of non-atomic agents. We assume m (H\A) ¹ 0. Then if every a Î A has the same preference-endowment pair, (>_a, e(a))_{aÎ A} where >_a are, additionally, convex. Then the set of core allocations is equal to set of Walrasian equilibrium allocations, C(E) = W(E).

Proof: Originally, Shitovitz (1973). We follow Greenberg and Shitovitz (1986) step-by-step.

Without loss of generality, assume that the atomless part is H\A = [0, 1]. Define a new economy (H*, Á *, m *), where H* = [0, 1 + m (A)] (atomless part plus the measure of all the atoms). Á * and m * are obtained as follows. Let ([1, 1+m (A)], J , l ) be an atomless Lebesgue measure space, thus [1, 1+m (A)] is the space of "atoms" substituted by atomless agents; J is the s -algebra on this segment and l is the Lebesgue measure. Now, Á * = Á ₁ È J , where Á ₁ is Á restricted to H\A = [0, 1], i.e. if S Î Á , then S Ç [0, 1] Î Á * and J is the s -algebra on [1, 1+m (A)]. Similarly, m *(S) = m (S) if S Î Á ₁ while m *(S) = l (S) if S Î J .

In short, (H*, Á *, m *) is a measure space which "replicates" the original economy (H, Á , m ), but with a set of atomless agents [1, 1+m (A)] "replacing" the original set of atoms. For intuition, see Figure 2. We shall call a "split" of a particular atom a Î A as a*, so m (a) = m (a*). Every agent h Î a* is of the same preference-endowment type as the original atom, a. H* is the set of agents when all atoms a Î A are split and added to the atomless part H\A. We shall let H₁* = H\A and H₂* = [1, 1+m (A)], so H* = H₁* + H₂*.

Now, allocations are defined as mappings from the space of names to the commodity space Rⁿ. As we have two name spaces, we can define two types of allocations, x: H ® Rⁿ and x*: H* ® Rⁿ. To translate an allocation x to x* the rule is:

if h Î H₁*, then x*(h) = x(h)

if h Î a* Ì H₂*, then x*(h) = x(a), where a Î A.

To translate from x* to x:

if h Î H\A, then x(h) = x*(h)

if a Î A, then x(a) = [ò _A* x*(h) dm *]/m *(a*) where h Î a* Ì H₂*.

Now, we want to prove that the cores of both economies are identical, i.e. defining C(E) and C(E*), where E is the original economy and E* is the new economy, then the cores are called "equivalent" if:

(i) x* Î C(E*) Þ x Î C(E)

(ii) x Î C(E) Þ x* Î C(E*)

where x and x* are related to each other as noted above. It is easy to split atoms -- so whatever is blocked in H, can be blocked in H*, but it is a bit trickier to "reconstitute" atoms and thus ensure the reverse, so (ii) is tougher.

(i) Suppose x* Î C(E*) but x Ï C(E). Then, there exists a coalition S Î Á and an allocation y such that x is blocked. But then the equivalent coalition, S* Î Á * (using the splits, the a*, of whatever atoms may have existed in S) can block the corresponding x* via the corresponding y*. Thus x* Ï C(E*). A contradiction.

(ii) Suppose x Î C(E) but x* Ï C(E), then x* can be blocked. Now, recall by Vind (1972) that, in an atomless space, if a coalition can block an allocation, then we can find a coalition of any smaller size that can block it too. Consequently, let us take the worst-served atom in H, i.e. find a₀ Î A where x(a) ³ _h x(a₀) for all a Î A. Let us define a as its measure, a = m (a₀). Now, by Vind (1972), consider a coalition S* Î Á * with size m *(S*) = 1+a ₀ which can block x* with allocation y*. Now, 1 + a = m *(S*) = m *(H₁* Ç S*) + m *(H₂* Ç S*). We also know that m *(H₁* Ç S*) £ m *(H₁*) = 1 by construction. Thus, necessarily, m *(H₂* Ç S*) ³ a . Now, by Vind (1972), if S* blocks the allocation x* with y*, so can a smaller coalition. Thus, by a slight extension of that argument, we can find an R* Ì S* such that m *(H₂* Ç R*) = a , which blocks x* via y*. Intuitively, we block with a coalition R* composed of both regular atomless folks and "döppelgangers" of the split atoms -- where the latter, together, are of size a .

Now, going back to the original economy, let us define coalition R as:

R = (H₁ Ç R*) È a₀

(recall H₁ is the set of atomless traders, so H₁ Ç R* are all atomless). Thus, this coalition is composed of the worst-served atom, a₀, and those atomless traders who had blocked things in the completely atomless space H*. Now, consider these first. If h Î H₁ Ç R, then y(h) = y*(h) and x(h) = x*(h) by definition of the translation. So, if in the atomless space, y*(h) >_h x*(h) by all h Î H₁ Ç R*, then it will also be true in the original space, i.e. y(h) >_h x(h).

What about the atom, a₀? In principle, recall, the atom a₀'s doppelangers in the atomless space are a₀*. Consequently, he ought to receive by translation:

y(a₀) = [ò _a°* y*(h) dm *]/m *(a₀*)

However, the folks in a₀* are not necessarily the same people in (H₂* Ç R*), name for name. But, by construction, there are enough people of that type in the atomless space who joined the blocking coalition which, by construction, together have the same size as the a₀ atom (as, recall, m *(H₂* Ç R*) = a = m (a₀)). Consequently, atom a₀ can "reconstruct" himself via these coalition participants. Thus, with no loss of generality:

y(a₀) = [ò _{H2* Ç R*} y*(h) dm *]/m *(H₂* Ç R*)

= [ò _{H2* Ç R*} y*(h) dm *]/a

All that remains to be shown is that y(a₀) >_h x(a₀). Now, note that ò _{H2* Ç R*} y*(h) dm * is the average of what H₂* Ç R* received from the blocking coalition in the atomless space. Now, they preferred y*(h) to x*(h). But recall by definition, x*(h) = x(a) for all h Î a₀*, which is what atom a₀ would get otherwise. So, if his representatives (all of which joined the coalition by construction) forsaked x*(h) for y*(h), then by convexity of a₀'s preferences, he ought to prefer the average of what his representatives got. Consequently, in combination with the atomless co-conspirators in the original space, H₁ Ç R*, the atom a₀ will move to block x with y. It is, by construction, feasible. Thus, (ii) is proved.

So now we approach the final part of the argument, proving that the core of the atom economy is equal to the set of Walrasian equilibrium allocations, i.e. C(E) = W(E). By Aumann's (1964) theorem, we know for the atomless counterpart that C(E*) = W(E*). All that remains is to show that if x* Î W(E*), then x Î W(E). To see this, note that by definition, if x* Î W(E*), then almost everywhere in H*, x*(h) is a maximal element with respect to the budget constraints of the agents. For the regular atomless agents, h Î H₁, this will be the same allocation in both economies, i.e. x(h) = x*(h). Now, as all of atoms's doppelgangers, i.e. all h Î a* for all a Î A, have the same same preferences, endowments and face the same prices, so they all receive the same allocation, x*(h). By translation, the atom a Î A gets x(a) = ò _a* x*(h)/m *(a*) for all h Î a*. But by the same argument, all the atoms have the same preferences and same endowment (scaled for size) and face the same prices. Thus x(a) will also be a maximal element with respect to their budget constraints. Consequently, x Î W(E).

In sum, it is elementary to prove that if x Î W(E), then x Î C(E). And we have just proved the chain x Î C(E) Þ x* Î C(E*) Þ x* Î W(E*) Þ x Î W(E). Thus, C(E) = W(E). And we are done.§