Edgeworth's Conjecture

A Blocking Coalition


"[W]e see that in general for any number short of the practically infinite (if such a term be allowed), there is a finite length of the contract-curve...at any point of which if the system is placed, it cannot by contract or recontract be displaced; that there are an indefinite number of final settlements, a quantity continually diminishing as we approach a perfect market."

(Francis Ysidro Edgeworth, Mathematical Psychics, 1881: p.38-9)

哲ow the rationale for this deduction, the reason why the complex play of competition tends to a simple uniform result - what is arbitrary and indeterminate in contract between individuals becoming extinct in the jostle of competition -- is to be sought in a principle which pervades all mathematics, the principle of limit, or law of great numbers as it might perhaps be called.・nbsp;

(Francis Ysidro Edgeworth, "The Rationale of Exchange", Journal of Statistical Society of London, 1884: p.164)

"This book shows clear signs of genius, and is a promise of great things to come...It will be interesting, in particular, to see how far he succeeds in preventing his mathematics from running away with him out of sight of the actual facts of economics."

(Alfred Marshall, 1881, "Review of Edgeworth's Mathematical Psychics", The Academy)

"It is not the mathematics which is running away with Edgeworth but his love for the classics. Edgeworth's arguments satisfy today's standard of rigor."

(Werner Hildenbrand, 1993, "Francis Ysidro Edgeworth", European Economic Review)



(1) Competition and the Core
(2) Replication
(3) The Shrinking Core
(4) The Limit Theorem

(1) Competition and the Core

Having denied Marshall's argument that barter exchange would, under certain circumstances, lead necessarily to the Walrasian equilibrium, Edgeworth offered up a far more interesting conjecture about the relationship between his core and the Walrasian equilibrium.

Edgeworth established that barter exchange yields the indeterminacy known as the core. Naturally, the easiest way to restore determinacy in barter exchange would be to use market prices as guides in the Walrasian manner. This would cut through the whole deadlocked mess and let the agents jump directly to the equilibrium. But the Walrasian mechanism works only if people agree to accept the prices as "given" and state their "true" demands. Why should they obey it?

The traditional answers, as we have noted elsewhere, have been (i) the Austrian argument: prices are efficient -- they convey all the information that is "needed" and thus people will naturally opt to use them as guides; (ii) the Paretian argument: the world is such a complicated predictive mess and competition is so intense, that one cannot hope to profit by playing strategically with prices; people have no choice but to take prices as given.

Edgeworth offered up another, even more subtle argument. Firstly, he asserted, people never take prices as given, they always work on the principle of recontracting. Secondly, under conditions of perfect competition -- i.e. an infinite number of people -- the results yielded by the recontracting process are identical to those yield by a Walrasian price-guided process.

It is important to stress Edgeworth's perspective on this point. It is not that perfect competition makes people price-takers (as Cournot and the Paretians might suggest), but rather it is merely that under perfect competition, the results are the same. According to Edgeworth, people are never price-takers, they are never guided by prices. However, if one wants to use Walrasian theory in one's analysis, one can. But Edgeworth's point is not that Walrasian analysis is correct -- it never is -- but rather that it is a way of finding the answer quickly. It merely summarizes the results of the more complicated but true underlying trading process of Edgeworthian recontracting. But even this "practical" use of Walrasian theory, he is careful to emphasize again and again, is only justified under conditions of perfect competition.

Here we must remind ourselves of what Edgeworth meant by "competition". This, as we noted, was "sheer numbers of people" -- nothing to do with prices. Perfect competition is the limit case of this -- an infinite number of people. More explicitly, competition is perfect if "any individidual is free to recontract with any out of an indefinite number [of individuals]" (Edgeworth, 1881: p.18) -- where, he notes on the same page, is analagous to there being an "infinity" of traders.

[Note: Edgeworth's obsession with the "numbers" theory of competition can be partly ascribed to his interest in statistics. Edgeworth (1884) drew a rather interesting analogy between his own "limit theorem" on core convergence to equilibrium with the Central Limit Theorem of statistics. There is a very revealing detail in this analogy: the central limit theorem claims that the relative frequency in the limit will be the same as the true probability, although the true probability is not defined as the relative frequency in the limit (in Edgeworth's time; later on, it became so defined). Similarly, the Edgeworthian core in the limit is the same as the Walrasian equilibrium, but the Walrasian equilibrium is not defined as the core in the limit. Thus, as we have asserted, Edgeworth is nowhere trying to justify Walrasian "price-taking" as what happens in the limiting core, but rather merely establishing a correspondence of outcomes.]

What Edgeworth set out to prove has become known as Edgeworth's conjecture. This is the following:

"The problem to which attention is specially directed in this introductory summary is: How far contract is indeterminate...The general answer is -- (a) Contract without competition is indeterminate, (b) Contract with perfect competition is perfectly determinate, (g) Contract with more or less perfect competition is less or more indeterminate." (Edgeworth, 1881: p.20)

What is the measure of indeterminacy? The "size" of the core, i.e. the number of allocations in the core. It is important to note that when stating this conjecture, Edgeworth was assuming the Walrasian equilibrium was unique -- thus he reference in (b ) to contract being "completely determinate". So let us restate Edgeworth's conjecture in its three parts into their modern form:

(a ) that if competition is anything less than perfect, then the core will be "larger" than the set of Walrasian equilibrium allocations;

(b ) that if there are an infinite number of people, then the only allocations in the core will be the Walrasian equilibria;

(g ) that as we increase the number of people in an economy, the number of allocations in the core is reduced.

(2) Replication

With surprising virtuosity, Edgeworth provided much of the proof of his conjectures himself. His suggestion was simple, as he notes:

"But it is not the purport of the present study to attempt a detailed, much less a polemical, discussion of pure Catallactics, but rather (g ) to inquire how far contract is determinate in cases of imperfect competition. It is not necessary for this purpose to attack the general problem of Contract qualified by Competition, which is much more difficult than the general problem of unqualified contract already treated. It is not necessary to resolve analytically the composite mechanism of a competitive field. It will suffice to proceed synthetically, observing in a simple typical case the effect of continually introducing into the field additional competitors." (Edgeworth, 1881: p.34)

Translated into clearer English, Edgeworth is suggesting that instead of trying to prove the limit case that, with an infinite number of agents, the core is identical to the set of Walrasian equilibria ("the general problem of Contract qualified by Competition"), it is sufficient to just prove that as we increase the number of agents, the core "shrinks", i.e. that the core that emerges when there are many traders is contained inside the core that emerges when there are only a few traders. Thus Edgeworth proposed that conjecture (g ) can serve as a substitute for conjecture (b ).

Edgeworth's procedure in proving this was effectively the same as that followed later by Herbert E. Scarf (1962) and Gérard Debreu and H.E. Scarf (1963): namely, by the replication of the economy. In other words, the introduction of more agents is done by a process whereby the agents that are introduced have identical preferences and endowments and brought into the economy in the same proportion as the ones already there. To a Robinson and a Friday, we add n Robinsons and n Fridays. This was explicitly proposed by Edgeworth:

"let us now introduce a second X [Robinson] and a second Y [ Friday]; so that the field of competition consists of two Xs and two Ys. And for the sake of illustration (not of argument) let us suppose that the new X has the same requirements, the same nature as the old X; and similarly that the new Y is equal-natured with the old." (Edgeworth, 1881: p.35).

Why replicate? Why add another Robinson and another Friday? Why not just introduce a third and fourth person, say Quixote and Sancho, with different preferences, endowments, etc.? After all, as Edgeworth himself notes, "the theorem admits of being extended to the general case of unequal numbers and natures" (1881: p.43).

The answer is that replication is the easiest way to maintain comparability. Adding new people with new preferences and endowments, changes the dimensionality of the economy and will make the new core more difficult to compare with the old core. With replication, we can maintain an economy with n Robinsons and n Fridays, without ever having to come out of the simple two-dimensional Edgeworth-Bowley Box. We can then actually "see" the core shrinking. If we were to add a Quixote and a Sancho to Robinson and Friday, then we would have to make the box four-dimensional. To verify whether the core "shrunk" or not , we would then have to compare the new core of the four-dimensional box with the old core of the two-dimensional box. That's not the easiest thing in the world to prove -- and much mathematically-intensive labor went into constructing such a proof throughout the 1960s and 1970s. Edgeworth may have been a good mathematician -- but not that good.

The crucial underlying notion that maintains comparability is the famous result known as "equal treatment in the core". Effectively, this means that agents of the same type will, in a core allocation, receive the same allocation. This allows us to represent an economy with n Robinsons and n Fridays in a two-person Edgeworth box diagram. To see why, it is best to start as Edgeworth did: namely, going from a two-person to a four-person economy -- with two Robinsons and two Fridays. Now, this four-person economy can be represented in a two-person box diagram (Figure 1) by simply overlaying the indifference map of Robinson #2 on Robinson #1 -- so they both start at the same origin (OR1 and OR2 are the same) and their endowments and indifference maps are identical. The same applies to Friday #2, who we overlay on Friday #1's map and so both Fridays start from origin OF1, OF2.

Now, we claimed that agents of the same type will receive the same allocation in the core. To see this result, firstly note that a pair of agents of the same type will not trade with each other: there are no gains from trade for Robinson #1 to trade with Robinson #2 because they have identical endowments and preferences. Now, suppose we "pair off" Robinsons and Fridays, e.g. Robinson #1 trades with Friday #1 and Robinson #2 trades with Friday #2. Suppose, as a result, that Robinson #1 and Friday #1 go to allocation A in Figure 1, while Robinson #2 and Friday #2 go to allocation B. This, of course, is supposed to be a multi-lateral trade involving all four agents, but we can decompose it into a pair of trades between opposing partners, as that permits representation in the two-dimensional box in Figure 1.

Notice that with this (A, B) allocation, the Robinsons receive different utility levels -- Robinson #1, at UR1, being considerably better off than Robinson #2 at UR2. Similarly, Friday #2 is much better off than Friday #1. Now, we assert, this allocation is not in the core. To see why, notice that the worse-off agents -- Friday #1 and Robinson #2 -- can pull out of the economy and trade with each other alone and achieve an allocation such as C in Figure 1. Here Robinson #2 receives UR2¢ and Friday #1 receives UF1¢ -- considerably better than their original allocations. Thus, the coalition of Friday #1 and Robinson #2 has "blocked" the originally proposed (A, B) allocation. Consequently, it cannot be in the core.

Fig. 1 - Replication and Equal Treatment in the Core

The lesson from this, then, is that there must be equal treatment in the core: if an allocation is to be a core allocation, then both Robinsons must receive the same allocation, and both Fridays must receive the same allocation. So, if we proposed again to pair-off Robinson #1 and Friday #1 and they achieve A in Figure 1, then we must also ensure that Robinson #2 and Friday #2 will also achieve A. If so, then the allocation is in the core because the relatively "worse off" parties (in this case, Friday #1 and Friday #2), cannot pull out of the economy and trade with each other (recall: there are no gains from trade between agents of the same type). So if everyone goes to A, then A is in the core. Similarly, if everyone goes to B, then B will be an allocation in the core. The same applies if everyone goes to C. But, in the core, we cannot treat agents of the same type differently.

We should note that this equal treatment in the core property was anticipated by Edgeworth. As he writes, quite clearly:

"Then it is evident that there cannot be equilibrium [i.e. a core allocation] unless (1) all the field is collected at one point; (2) that point is on the contract curve. For (1) if possible let one couple be at one point, and another couple at another point. It will generally be the interest of the X of one couple and the Y of the other to rush together, leaving their partners in the lurch. And (2) if the common point is not on the contract-curve, it will be the interest of all parties to descend to the contract curve." (Edgeworth, 1881: p.35)

This is what makes two-dimensional representation of an economy with n agents of two types possible.

(3) The Shrinking Core

So let us now turn to the shrinking core. If we only had one Robinson and one Friday, then the core would be the portion of the contract curve stretching from A to B in Figure 2. What will happen when we introduce another Robinson and Friday? We claim the core will "shrink".

Fig. 2 - Blocking Old Core Allocations

To see why, focus on allocation B in Figure 2. In the two-person economy, it is in the core, so if Friday proposes allocation B, Robinson cannot block it. However, in a four-person economy, things are different. Supposed one of the Fridays (say Friday #1) proposes allocation B, i.e. proposes that the each of the Robinsons gives up D x in return for D y, while each of the Fridays surrenders D y for D x. Then each of the Fridays will achieve utility UF¢ , while each of the Robinsons only achieve utility UR.

Can this be blocked? It can. A coalition can be formed between both of the Robinsons and the remaining Friday (Friday #2) that blocks the allocation B. To see why, note that the Robinsons can propose the following to Friday #2. Robinson #1 speaks:

"Forget Friday #1! Let us trade among ourselves; me and Robinson #2 will each give you D x¢ , so you receive a total of D x = D x¢ + D x¢ (which is the same as you would have received at B), while you give us each D y¢ (so you pay a total of D y = D y¢ + D y¢ , the same as you would at B). Doing so, you will receive the same utility in our coalition, UF¢ , as you would under Friday #1's original proposition. However, we will be better off! Because I only give up D x¢ and receive D y¢ , I'll be at position F (the mid-point of the chord between endowment and B in Figure 2) -- thus enjoying utility UR¢ , which is much better than the UR I'd receive under Friday #1 proposition. Robinson #2, who also gives up D x¢ and receives D y¢ , will also be at F and enjoying the same utility, UR¢ . Thus, me and Robinson #2 will be better off, while you are no worse off. So, in sum, you will be at B, each of us will be at F, while we shut out Friday #1, who will then have to remain at his own endowment point, e and achieves only utility UF. Are we agreed?"

Friday #2 would have no reason to reject the coalition offer: he does just as well under it as he does under Friday #1s. Robinson #1 and Robinson #2, of course, will participate because they are strictly better off. So, Friday #1's proposition of allocation B for everybody is "blocked" by a coalition of two Robinsons and one Friday. Thus the allocation B is no longer in the core.

It is elementary to note, of course, that given that they can achieve at least allocation F by forming a coalition with Friday #2, the Robinsons will reject any trade that takes them to an allocation that is below the utility level UR¢ . Thus, all the old core allocations between B and C will now be "blockable" and thus no longer part of the new core. Edgeworth, once again, had spelled this out explicitly:

"the points of the contract-curve in the immediate neighborhood of the limits [i.e. autarky indifference curves]...cannot be final settlements. For if the system be placed at such a point...it will in general be possible for one of the Ys (without the consent of the other) to recontract with the two Xs, so that for all three parties the recontract is more advantageous than the previously existing contract." (Edgeworth, 1881: p.35)

Of course, the Fridays for their part can also do the same. If one of the Robinsons (say Robinson #1) offers allocation A in Figure 2, which was previously part of the core, then both of the Fridays together with the remaining Robinson #2 can form a coalition to block it: Robinson #2 being taken to point A, while each of the Fridays go to allocation G (the mid-point of the chord connecting endowment and A in Figure 2) and achieve utility UF¢ ¢ -- much better than they would receive under Robinson #1's proposed A. Thus, the Fridays can block all allocations which give them less utility than UF¢ ¢ -- which includes all of the old core allocations between points D and A.

Now, it is important to note that the new line segment, CD in Figure 1, is not the new core. This is because it might also possible to block C itself as well as other points in the CD segment. Consider Figure 3, where have now added a chord connecting the endowment point and allocation C. The mid-point of this chord is J, thus two Robinsons and one Friday can again block C by proposing to Friday #2 to take him to C, while they split the trade among themselves and acheve J each -- with considerably higher utility than what they had at C -- while leaving Friday #1 at his endowment. Thus, the coalition will also be able to block any old core allocation between K and C.

Figure 3 - More Blocking

However, note that the allocation N in Figure 3 is not blockable by our two Robinson-one Friday coalition. This is because if Friday is to be placed at N and the Robinsons split the trades that put him there, then each of the Robinsons will end up at M -- which yields a utility level that is below UR¢ ¢ ¢ , the utility they would receive at N. So, N is not blockable by our coalition. N is in the new core.

It is interesting to note, however, that if we add more agents, then points such as N can be blocked. To see this, examine Figure 4, where we have N again. The point M is the mid-point on the chord connecting endowment and N -- and this represents what we would obtain with a two Robinson-one Friday coalition, which would not block N. But now suppose that, by replication, we increase the number of agents to six (three Robinsons, three Fridays). It is easy to see that the previously unblockable core allocation N is now blockable.

Fig. 4 - Blocking with Even More Agents.

To see this, note that a coalition of three Robinsons and two Fridays can block N with allocation Q (leaving the third Friday out in the cold). Specifically, Friday #1 and Friday #2 will each go to N and receive utility UF¢ . So, the coalition of three Robinson must come up with 2D x to pay for the 2D y that the Fridays have given them. They'll split the bill: each Robinson contributing D x¢ = (2/3)D x and receiving D y¢ = (2/3)D y. Thus, each of the three Robinsons will be at allocation Q in Figure 4, which is two-thirds of the way up the chord from the endowment to N, and each will receive the higher utility UR¢ ¢ . Allocation N is now blocked.

Thus, a coalition of three Robinsons and two Fridays can do now what a coalition of two Robinsons and one Friday could not do before. As whatever two Robinsons and one Friday could block in the four-person economy, they will be able to continue to block in the six-person economy, it consequently follows that the core in a six-person economy is smaller than the core in the four-person economy. So, the core shrinks further.

In sum, increasing the number of agents via replication, we are guaranteed that the core shrinks as the coalitional possibilities are increased. Part (g ) of Edgeworth's conjecture seems to be right: more people means a smaller core.

(4) The Limit Theorem

What about part (b)? This is the famous limit theorem: namely, increasing the number of people to infinity, leads the core to shrink completely so that the only allocations that are left are the Walrasian equilibria. A more general and formal proof of this, for a replica economy, was provided a century later by Gérard Debreu and Herbert E. Scarf (1963) -- and for more general economies, it took even longer.

Nonetheless, the basic steps of the proof were already in Edgeworth (1881: p.35-39). It might be worthwhile, then, to provide here a restatement of his analysis in Edgeworth-Bowley boxes.

[Note: although Edgeworth's reasoning is effectively the same as what follows, his garbled English obscures it. Werner Hildenbrand (1993) deciphers the theorem from a careful reading of Edgeworth's text and diagrams. A simple calculus-based version of this is found in Leif Johansen (1978).]

Consider Figure 5, where we have to allocations, C and D. Obviously, C is an equilibrium allocation, while D, which is in the core, is not an equilibrium allocation. The basic proposition is that we cannot block equilibrium point C with any sized economy, while we will eventually be able to block a point such as D with a sufficiently large number of (replicated) agents.

This result makes sense when examining the chord between the endowment allocation, e, and the allocation of interest (in this case C or D). We see immediately that at D, the utility level achieved by each of the Fridays is UF¢ while the utility achieved by each of the Robinsons is UR¢ . Now, consider the chord that goes from e to D. Obviously, allocation G along this chord is better for the Robinsons than D -- as they now achieve a higher utility level UR¢ ¢ . Now, like the case that went before, they would like to form a coalition with some of the Fridays so that all the Robinsons go to G, while some of the Fridays go to D, while the remaining Fridays stay at their endowment point. This would effectively block D as an economy-wide allocation.

Now, not just any coalition will do. A small economy may not be able to block D. We need a sufficiently large economy. This is because, as we saw in Figure 4 above, with more agents, we are able to move the coalitional allocation each Robinson receives up the endowment-allocation chord towards the allocation D. So, as long as the endowment-allocation chord cuts the Robinsons' indifference curve UR¢ in the interior, we will always be able to find an interior point (such as G) in a sufficiently large replica economy. Thus, as we increase the number of agents, we will eventually be able to block any non-equilibrium core allocation point such as D. But the Walrasian equilibrium can never be blocked by any-sized replica because the chord connecting C and the endowment does not cut either of the indifference curves through the interior.

Fig. 5 - Core and Equilibrium

So let us begin. Let us first remind ourselves that points D, G and e are just allocation points. They can be decomposed into the parts everyone receives. So, let eR be the endowment of one Robinson, while eF is the endowment of one Friday. Also, let DR be the allocation that goes to one Robinson and DF the allocation that goes to one Friday if the proposed allocation D is realized. Finally, let GR denote the allocation that goes to one Robinson if they manage to attain the allocation G.

Now, suppose that our economy is an n-fold replica (i.e. n Fridays and n Robinsons). The proposed coalition is thus made up of n Robinsons and (n-r) Fridays (and so the number of remaining Fridays is r). Now, under the coalitional allocation, each of the n Robinsons is to receive GR, while each of the coalition Fridays are to receive DF. The remaining Fridays are shut out, and thus receive only their endowment, eF.

Now, as G is above UR¢ , then obviously all the Robinsons strictly prefer allocation GR to allocation DR. The Fridays in the coalition receive what they would have received otherwise, DF, so they are indifferent. So it is clear that a coalition of n Robinsons and (n-r) Fridays will lead the Robinsons to be better off and the participating Fridays no worse off. The only thing remaining to ensure that D is blocked is is to check that this coalition is feasible, i.e. that the endowments of the participants are enough to fulfill the necessary coalitional trades. Now, the total amount that is needed for the coalition to work is nGR + (n-r)DF. The total endowments of the coalition are neR + (n-r)eF. So we need to check that nGR + (n-r)DF £ neR + (n-r)eF.

To see that this is true, notice that in Figure 5, G lies on the chord connecting allocation D and the endowment, e. Thus, for each of the Robinsons, we can write:

GR = l eR + (1-l )DR

where l is some fraction, so GR is some convex combination of their endowment (eR) and the allocation DR they would have received if D was realized. Now, let us propose l = r/n (it is a fraction, since r < n, because the remaining Fridays, r, are always less than the total number of Fridays, n). Thus:

GR = (r/n)eR + (1-(r/n))DR

Now, notice that we can plug this into our total coalition demand and obtain:

nGR + (n-r)DF = n[(r/n)eR + (1-(r/n))DR] + (n-r)DF


nGR + (n-r)DF = reR + nDR - rDR + (n-r)DF

= reR + (n-r)(DR + DF)

Now, as the old allocation D was feasible by definition, then n(DR + DF) = n(eR + eF) or, dividing by n, DR + DF = eR + eF. Thus inserting this:

nGR + (n-r)DF = reR + (n-r)(eR + eF)

or simply:

nGR + (n-r)DF = neR + (n-r)eF

which is exactly the coalitions' endowment -- thus the coalitional allocation is feasible. In other words, point D is blockable by a coalition of n Robinsons receiving GR and (n-r) Fridays receiving DF.

There are two tricks to this story: firstly, it is in finding the appropriate l = r/n, but as any n-fold replica is allowed and any coalition can be formed, this is completely flexible; the second trick is being able to find an allocation such as G which some members (in this case the Robinsons) will strictly prefer to the originally proposed allocation D. This will be true for any allocation in the core that is not a Walrasian equilibrium (albeit provided that the assumption of monotonicity and strict convexity of preferences remains fulfilled).

Why? Well notice in Figure 5 that the chord connecting the Walrasian equilibrium C to the endowment point passes between the indifference curves at C. This chord, of course, is the equilibrium price line. But the main lesson is that as this chord does not cut through the interior of either of their indifference curves at C, then there is no point on this chord which either the Robinsons or the Fridays prefer strictly to what they receive at C. Thus, it is impossible to find a coalition that will block the Walrasian equilibrium -- regardless of the size of the replication.

In sum, this diagrammatic sketch effectively shows two results: (1) that every core allocation that is not a Walrasian equilibrium can be blocked with a sufficiently large replica economy and (2) that no Walrasian equilibrium can ever be blocked. So, if we have an infinite number of agents of each type (an infinite replica), then by (1), every non-equilibrium allocation will be blocked out of the core and, by (2), every Walrasian equilibrium allocation will not be blocked. This is the heart of the "limit theorem", Edgeworth's proposition (b): with an infinite number of agents, the set of core allocations is identical to the set of Walrasian equilibrium allocations.

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