A market too large to bargain in

A split nobody can walk away from began with the core: the set of outcomes that no group of players can improve on by acting alone. In the glove market the core collapsed to a single point as soon as one glove was in surplus — the market’s price, with nothing left to bargain over — and it did so with three players. That collapse was a special feature of gloves, which are perfect substitutes for each other. In a market for ordinary goods, two traders can strike many different deals, and the core is a whole range of them.

The question Francis Ysidro Edgeworth asked in 1881 is what happens to that range as the market grows. His answer, argued with diagrams and proved for the case of a few kinds of trader eighty years later, is that it shrinks — that adding traders turns bargaining into price-taking, because larger and larger groups gain the power to refuse any deal that is not close to what prices would give. This essay computes the shrinking for one small economy, and shows which groups do the refusing.

Two traders and a box of deals #

Two goods, call them bread and cloth. Trader A owns 3 units of bread and 1 of cloth; trader B owns 1 and 3. Each values a bundle by the product of its two amounts — a standard way to say that both goods matter and that a balanced bundle beats a lopsided one. Every way to divide the 4 units of each good between them is a point in a square, measured from A’s corner for A’s share and from the opposite corner for B’s.

The efficient divisions — those where neither can be made better off without making the other worse off — lie along the diagonal, where each trader holds bread and cloth in equal amounts. A trade that leaves either trader worse off than their own endowment will be refused, so only part of the diagonal is acceptable to both: the part between the two curves through the endowment. A’s share of each good there runs from $\sqrt3 \approx 1.732$ to $4 - \sqrt3 \approx 2.268$. Every point of that stretch is in the core of the two-trader market. Bargaining power, patience, or luck decides where on it they settle.

Edgeworth drew this box himself, in Mathematical Psychics of 1881, and named the diagonal the contract curve: the deals two traders who have finished bargaining might sign. His point was that the contract curve is not a price. Two traders, or two trade unions facing two employers’ associations, have a range of mutually acceptable terms, and economics alone does not say where in the range they land. What a larger market adds, he argued, is recontracting: any deal can be torn up if a group of traders finds better terms among themselves, and with more traders there are more groups that might. He believed the process ended at the price as the market grew without limit, and he gave the argument for small numbers of traders by hand. The general proof waited until 1963.

One point on the stretch is special. If bread and cloth trade at one for one, A can sell bread for cloth at that rate, and A’s best bundle at that price is (2, 2); B’s is (2, 2) as well, and the two plans fit together exactly. That is the competitive equilibrium — the outcome a market with a posted price produces — and it lies in the middle of the core. In a two-trader market, nothing singles it out: it is one acceptable deal among a continuum.

Two of each, and a group of three #

Now replicate the economy: two traders exactly like A, two exactly like B. The deals to consider are those that give each A the share $t$ along the diagonal and each B the rest — and any group of traders may refuse a deal if, pooling only its own endowments, it can make every member strictly better off.

A deal giving each A the bundle (2.15, 2.15) is in the core of the two-trader market — it is between the curves. In the four-trader market it is not. One A and two B’s can pool their goods, (3, 1) + (1, 3) + (1, 3), and divide them so that the A gets exactly as much as the deal promised and each B gets more. The extra B is what makes it work: with two B’s competing for one A’s bread, the A can be offered a bundle the A likes as much as the deal while the B’s keep more for themselves. In the two-trader market there was no second B to compete, and the deal stood.

Each replication adds groups of new shapes, and each shape can refuse deals the smaller market had to accept. That is the mechanism. Competition in Edgeworth’s sense is not a price being announced; it is the presence of other traders who can be invited into a group.

Treating equals equally #

A deal in the replicated market could give the two A’s different bundles. It never survives.

Pair the worst-treated A with the worst-treated B. Their own endowments together are the whole economy’s in miniature, and dividing them as the average deal would — which, with preferences like these, both like at least as much as their own share — gives each of them at least the average, which is more than the worst-treated were getting. So any deal that treats traders of one kind unequally is refused by its two least favoured traders. Debreu and Scarf proved this equal treatment property in general for replicated markets with preferences of this kind; it is what allows the core to be drawn, as in the first figure, as a single share $t$ on the diagonal rather than a separate bundle for every trader.

Why the core is never empty here #

A core can be empty. In the game where any two of three can take a pound, every division is refused by some pair, and the question becomes which division makes the loudest objection quietest. Whether a game’s core is empty is decided by weighing families of coalitions against the whole: the core is non-empty exactly when no balanced family of coalitions can together claim more than the grand coalition can produce.

An exchange economy like this one always passes that test, and the reason is the price-equilibrium deal. At the equilibrium prices every trader is buying the best bundle their endowment affords, so any group that tried to do better among themselves would have to give some member a bundle that member could not afford at those prices — and adding up the members’ budgets, the group’s pooled endowment cannot afford all of their better bundles at once. So the competitive deal is in the core at every size, and the shrinking core always has something left in it. That the equilibrium exists in the first place is a fixed-point theorem: prices that clear every market at once are a point that a map from prices to prices leaves alone, the theme of the coffee that always has a molecule in place, and for preferences that are not smooth it needs the version of the theorem that allows a set of best replies.

Fifty of each #

With equal treatment established, the core of the market with $r$ traders of each kind is a stretch of the diagonal, and whether a given share $t$ survives comes down to whether some group of $a$ A’s and $b$ B’s, with $a$ and $b$ at most $r$, can refuse it. For these preferences the group’s best counter-offer keeps every bundle on the ray of its own pooled endowment, so the test is a short formula, and the surviving stretch can be computed for any $r$.

The range collapses quickly at first — from a width of $0.536$ to $0.168$ with the first replication — and then steadily. At ten traders of each kind A’s share is pinned between 1.987 and 2.013; at fifty, between 1.9975 and 2.0025. Every deal other than the competitive one is eventually refused, which is Edgeworth’s conjecture. And the price-equilibrium deal itself is never refused, at any size: no group, pooling its own goods, can beat what prices give each member, because each member is already getting the best bundle that member could buy with their own endowment at those prices.

How fast the room closes #

The width times the number of traders settles to a constant, about $0.25$, so the core shrinks in proportion to $1/r$. Gérard Debreu and Birgit Grodal showed in the 1970s that this is the typical rate for markets whose traders have smooth preferences, and the computation reproduces it with a definite constant for this economy. Doubling the market halves the room for bargaining.

The rate matters for how the theorem is read. Edgeworth’s conjecture is a limit, and limits can be slow; a $1/\sqrt r$ rate, for instance, would leave substantial room for bargaining in markets of hundreds. A $1/r$ rate means that a market of a few dozen traders of each kind already leaves almost nothing to bargain over: a deal can depart from the price by at most a fraction of a per cent before some group refuses it.

Who does the refusing #

The smallest group that refuses a given deal says something about where the pressure comes from.

The blocking coalitions always contain one more trader of the kind the deal shortchanges. A deal that favours A is refused by groups with one more B than A, and the closer the deal is to the price, the larger the group must be: one A with two B’s for deals far from the price, two with three, four with five, eleven with twelve within a hundredth of it. The B’s in such a group are competing with each other for the A’s goods, and the group’s ratio of B’s to A’s is a price in disguise — the more nearly equal the numbers, the more nearly the group’s internal terms of trade match one for one, and the more finely it can undercut a deal that departs from that ratio by only a little.

This is the picture that makes the limit theorem plausible. Prices are not imposed from outside; they are what is left when every group that could form has been allowed to form, and the groups needed to discipline a near-price deal are large because the deal is near the price. In a finite market there are only so many traders, and some departures survive. In the limit none do.

The other way to share a large market #

The core is one answer to how a market’s gains should be divided; another, met in an earlier essay, is to average what each trader adds over every order in which the traders might arrive. In small markets the two answers differ — in the glove market they differed sharply, the average over orders giving the short side a share the core gave entirely to the other. As the market grows they come together. Robert Aumann and others showed that for large markets with smooth preferences the averaged shares converge to competitive allocations, so the two theories, built on entirely different principles — one about what groups can refuse, the other about what each trader contributes on average — agree in the limit with each other and with the price. Three ideas of fairness that coincide only when no trader is large enough to matter is a strong argument that, in that limit, prices are not one convention among several but the only outcome with a claim to stability.

Gloves against bread and cloth #

The glove market, read against this one, shows what the general theorem depends on. Gloves are perfect substitutes within a side and perfect complements across — a left glove is worth nothing without a right — and in that market the core was already a single point with three players. Bread and cloth here are smooth substitutes, valued in combination, and the core shrinks gradually, like $1/r$. Between those extremes the rate depends on how easily traders can substitute one good for another, and a market with many kinds of trader and many goods has many such rates at once.

The prices for houses that no buyer or seller breaks away from are the same idea in a market where goods are indivisible: there the core is a lattice of price vectors with a best corner for each side, and it does not shrink, because each house is unique and no replication supplies a substitute for it. Shrinking needs competition among traders who are alike, and the indivisible market has none.

Two kinds of trader, fifty of each, one formula #

The computation is exact in its arithmetic and narrow in its scope. The test for whether a group refuses a deal is a closed formula for these particular preferences — products of the two amounts — and it relies on equal treatment within the group, which holds for these preferences and is checked for one unequal deal in the figure rather than proved in general there. The range for each $r$ is found by bisection on the formula to fifteen digits, checking every group of up to $r$ traders of each kind.

What the figures cannot show is the theorem in general: other preferences, more kinds of trader, more goods. Debreu and Scarf’s proof covers all of those for replicated economies, and Robert Aumann’s theorem of 1964 covers markets with a continuum of traders, each negligible, where the core and the competitive allocations coincide exactly. The constant 0.25 in the rate is a measurement for this economy; a different one would have a different constant.

Still open: markets that do not replicate #

Replication is a strong assumption. It supposes that the market grows by adding exact copies of the traders already present, so that every trader always has many identical rivals. Real markets grow unevenly, and their traders differ. For economies that grow in a general way — new traders of new kinds, in changing proportions — the core still converges to the competitive allocations under suitable conditions, a line of results due to Hildenbrand, Anderson and others; but the rates in that setting depend on how heterogeneous the traders are, and sharp rates are known only in special cases.

The other open edge is the one the glove market raised: what happens when goods are partly complements, so that preferences are not smooth. There the core can shrink in jumps or not at all, and which markets lie on which side is only partly understood: a structural condition on how goods substitute for one another — gross substitutability, in one form — is known to secure good behaviour in several related questions, but it is far from necessary, and markets that violate it can behave either way. A description of the rate at which bargaining disappears, in terms of how the goods in a market fit together, does not exist in general.

Refused by a large enough group #

Two traders with one deal to strike have a range of acceptable outcomes, and nothing in the economics picks one. Add identical traders and the range shrinks, because any deal that departs from what prices would give can be refused by some group of traders — larger the closer the deal is to the price. At fifty traders of each kind, the acceptable deals lie within an eighth of a per cent of the price. Competition in Edgeworth’s sense is the possibility of forming groups, and a market large enough has a group ready to refuse every deal except one.