One glove too many

A split nobody can walk away from defined the core of a cooperative game: the divisions of what the whole group earns that no smaller group could improve on by leaving and earning its own worth. It drew the core as what survives when every coalition’s threat cuts a line across the triangle of possible splits, and among its examples was a game of three players — two holding left gloves and one a right glove, a pair worth one and an odd glove worth nothing — in which the core shrank to a single point: everything to the right glove.

Three players is a small market. This essay takes the same game to markets of any size and finds that the single point is not an accident of smallness. Whenever one side of a glove market is short by even one glove, the core is a single split that gives the short side everything, and when the sides are exactly equal it is a whole interval of splits, one for every price between nought and one. The core jumps from one end of its range to the other when a single glove is added. The other classical rule, the average over orders of arrival, moves smoothly through the same change — and the contrast between the two is a contrast between two pictures of what a market is.

The game, and why a spare glove is worth nothing #

There are $L$ owners of left gloves and $R$ owners of right gloves, and a group of them is worth the number of pairs it can make: $v(S) = \min(\text{lefts in } S, \text{rights in } S)$. The whole market is worth $\min(L, R)$. A split gives each player a share of that total, and it is in the core if every group receives, in total, at least what it could make alone.

Suppose there are three lefts and four rights, so the market makes three pairs, and propose the split that gives each left $0.9$ and each right $0.075$. It shares out exactly three. Is anyone able to do better by leaving?

Yes. Take all three lefts and any three of the rights. They can make three pairs between them, worth $3$, and the proposed split pays them $3 \times 0.9 + 3 \times 0.075 = 2.925$. They are $0.075$ short of their own worth — exactly the share being paid to the right they left out — and so they reject the split. The same argument applies to any split that pays any right glove anything: leave out that right, pair everyone else, and pocket its share. The only split that survives pays the rights nothing and the lefts a whole pair each. It is in the core because no group can do better: a group with $\ell$ lefts and $r$ rights makes $\min(\ell, r) \le \ell$ pairs, and it already receives $\ell$.

The argument used nothing about the size of the market except that one side is longer. With a thousand lefts and a thousand and one rights, the same coalition — every left and all but one right — blocks any split paying any right a penny. A single spare glove, which nobody in particular owns, makes every right glove worthless.

Equal sides, and every price at once #

Now let the sides be equal. With five lefts and five rights there is no spare glove, and the blocking argument has nothing to leave out.

The core is now every split that gives each left the same amount $t$ and each right $1 - t$, for any $t$ between $0$ and $1$. Each pair is divided in the same proportion, because if some left received less than another, the underpaid left and the right partnered with the better-paid one would rather pair with each other. Every proportion survives, because any group makes exactly as many pairs as its shorter side has members and is paid at least that much. The core does not say what the price of a glove is. It says that any price clears the market when supply equals demand.

The figure sets the three markets side by side, and the jump is the whole point. Remove one right glove and the interval collapses to the point $1$; add one and it collapses to $0$. One glove moves the core from one end of its range to the other. No rule built from averages could behave like this; the core can because it is built from threats, and the only threat that matters is the one the surplus glove cannot answer.

The core as a market price #

Economists recognise the step at once: it is the textbook picture of supply and demand when both are perfectly inelastic. The lefts’ owners will sell at any positive price, and so will the rights’ owners; there is a fixed number of each. A competitive price for a left glove is one at which the number of pairs buyers want equals the number available. If lefts are scarce, every left will be bought at any price below the value of a pair and competition among the surplus rights drives the price of a right to nought, so a left fetches the whole pair. With equal numbers any split clears.

That the core and the competitive equilibrium agree is no coincidence. The glove market is the simplest case of the assignment game that the prices nobody can break away from studied — buyers and sellers paired off, each pair worth some amount — and there Lloyd Shapley and Martin Shubik showed in 1971 that the core is exactly the set of competitive prices, the solutions of the dual of the problem of pairing everyone up. In the glove market every possible pair is worth the same, the dual has the two corner solutions, and the core is their convex hull — a point or an interval.

It also settles a question that Francis Ysidro Edgeworth raised in 1881 about markets in general: does the core shrink to the competitive outcome only as the market grows large? Gérard Debreu and Herbert Scarf proved in 1963 that, for replicated exchange economies with smooth preferences, it does — the core contracts towards the competitive allocations as more traders of each type arrive. The glove market reaches the competitive outcome with no growth at all: three players already have a core that is a single competitive point. The reason is the perfect substitutability of one glove for another, which leaves a surplus trader nothing to bargain with.

The average over orders sees a lattice path #

The second rule asks something quite different. Line the players up in a random order; let each join in turn; pay each what it adds to the group already present; and average over all orders. The corners are the orders of arrival showed that each order’s payment is a corner of the core in a convex game. The glove game is not convex, and its average lands nowhere near its core.

What a left glove adds on arrival is easy to say: a pair, exactly when more rights than lefts are already present, and nothing otherwise. So draw an order of arrival as a path on a grid, a step up for each left and a step across for each right.

A left completes a pair exactly when its step starts below the diagonal; a right, when its step starts above it. Every path completes $\min(L, R)$ pairs, one for each time it moves away from the diagonal on the side that adds a pair, and a player’s value under the average over orders is the fraction of all paths on which its own step does so. The Shapley value of a glove is a count of lattice paths on one side of a diagonal — the same count as the ballot problem, which asks for the chance that one candidate stays ahead of another throughout the counting of votes, and which counting the paths that go wrong and the path folded at its first touch solved by reflection.

For one left and two rights the count gives $2/3$: the left is paid two thirds of a pair, the rights one sixth each. For two lefts and three rights it gives $13/20$. Both match a brute-force average over every order, which the figures check, along with the requirement that the shares add up to the number of pairs.

A large market with one glove to spare #

Now replicate the market: $n$ lefts against $n + 1$ rights, for $n$ from 1 to 200. The core never moves — every left gets the whole pair, every right nothing. The average over orders moves a great deal.

The warm values fall like $\tfrac12$ plus something of order $1/\sqrt n$ — the computed excess times $\sqrt n$ is $0.31$ at $n = 10$, $0.38$ at $n = 50$ and still creeping up at $n = 200$. In a market of four hundred and one glove owners, one glove short on the left, the average over orders pays each left $0.529$ of a pair and each right $0.469$. The scarce side is barely ahead. The ballot picture says why: in a long random path with one more step across than up, the path wanders on both sides of the diagonal, a fluctuation of order $\sqrt n$ swamps an imbalance of one, and a left arriving at a random moment finds itself below the diagonal only a little more than half the time.

The cool values tell the opposite story. With two lefts for every three rights the path has a drift — it moves across half as often again as it moves up — and the drift, which grows like $n$, beats the fluctuation, which grows like $\sqrt n$. Two hundred lefts against three hundred rights are paid $0.974$ of a pair each. When the shortage grows with the market, the average over orders comes round to the core’s answer; when the shortage is a fixed handful of gloves in a growing market, it goes to an even split instead.

Two rules that agree except on a knife-edge #

That pattern is a theorem in disguise. For markets with very many small traders, Robert Aumann showed in 1975 that the average over orders converges to a competitive allocation — the value equivalence principle — and Gérard Debreu and Herbert Scarf had shown in 1963 that the core shrinks to the competitive allocations too. In a glove market with a fixed proportion of lefts to rights the competitive allocation is unique and both rules reach it, as the cool line does.

A shortage of one glove is the borderline case. In the limit of a large market, one glove in four hundred is no shortage at all: the market looks balanced, its competitive prices form the whole interval from nought to one, and the average over orders settles at the interval’s middle. The finite market’s core, meanwhile, still sees the single spare glove and stays at an end. The two rules disagree exactly where the market is on the edge between two competitive answers, and there they disagree completely — one at an end of the interval, the other at its centre.

It also says something about which rule to trust in practice. Laboratory markets of the double-auction kind that Vernon Smith ran from the 1960s onwards tend, when trading is repeated, to move towards competitive prices, and a market with a large and persistent shortage behaves much as the core predicts. A market that is nearly balanced is where the core is least robust: its prediction jumps from one extreme to the other on the arrival of one trader, and a theory that sensitive to one trader is describing threats that real trading may not have time to exercise.

The other solutions take the core’s side #

The core is not the only rule built from objections, and the others agree with it here. The objection nobody can make louder defined the nucleolus, the split that makes the largest complaint of any coalition as small as possible, then the next largest, and so on; it always exists, and it lies in the core whenever the core is non-empty. With one glove short on the left the core is a single point, so the nucleolus is that point: everything to the lefts. In the three-player game of two lefts and one right, that essay found exactly this — the whole pair to the right glove — and five weighings and the question is closed certified that nothing else survives.

With equal sides the nucleolus has an interval to choose from, and it chooses the middle: each pair split evenly, the point where the complaints of the left-heavy and right-heavy coalitions are balanced. So on balanced markets the nucleolus and the average over orders agree, both at a half, and on unbalanced ones the nucleolus agrees with the core and the average does not. The pairwise objections of an objection one player makes to another lead to the same place for the same reason: a right glove objecting to a left can always be answered, because the left can name another right that would pair with it more cheaply.

This is worth noticing because the average over orders is the outlier, not the core. Every solution that is built from what coalitions can threaten gives the scarce side everything as soon as it is scarce; the one solution built from what players contribute moves smoothly. The divide is between the two families of rule, and the glove market is the cleanest place to see it — which is also why the four conditions characterising the average over orders say nothing about threats at all.

Which picture describes a real market #

Neither exactly, and the replica figure suggests why. A market with a substantial, persistent shortage behaves as both rules eventually predict: the scarce side captures most of the value, as the cool line does, and repeated laboratory trading drifts in that direction. A market that is nearly balanced is where the rules part, and it is also where the core is least robust as a prediction, because the threats it relies on — every coalition forming at no cost, every surplus trader instantly played off against the others — are the kind real trading has least time and least information to exercise. A theory whose answer jumps from one extreme to the other on the arrival of one trader is describing a limit of perfect competition, and a market of a few dozen people may sit well inside the interval instead.

What the pictures do not show #

The step figure looks like a function of $L$, and it is not quite one. At $L = R$ the core is an interval, and the bar drawn there is a whole set of answers, not a value; a reader should not interpolate across it. The ballot figure shows one path out of $\binom{21}{10} = 352{,}716$, and the Shapley value is an average over all of them; the single path illustrates the rule for who adds a pair, never the value.

The replica figure’s dashed curve, $\tfrac12 + 0.4/\sqrt n$, is a guide drawn to the eye. The computed values are exact sums, checked against enumeration for small markets, but whether the constant in front of $1/\sqrt n$ settles at $0.4$ or somewhat above — the computed ratio is still rising slowly at $n = 200$ — is not decided by eleven points on a logarithmic axis. And nothing here models time. A real market has an order of trades and a history; both rules are static descriptions of an outcome, and the question of which outcome a particular trading process reaches is a separate one with its own, much messier, answers.

Still open: how fast cores shrink when the market is not this simple #

For glove markets the core is computed exactly at every size, and nothing is open. For exchange economies in general the Debreu–Scarf theorem says the core shrinks to the competitive allocations as the economy is replicated, and later work by Birgit Grodal, Robert Anderson and others gave rates — for smooth economies the core’s distance from the competitive outcome falls roughly like $1/n$. What happens between the smooth case and the glove market, where the core is competitive from the start, depends on how far traders’ goods are substitutes or complements, and a description of the rate in terms of that degree of complementarity is not available in general.

There is a related question about the average over orders. Value equivalence holds for markets whose traders’ preferences are smooth, and in the glove market, which is as far from smooth as a market can be, it still holds whenever the competitive price is unique — the cool line in the replica figure. How far value equivalence extends to non-smooth markets in general, and how fast the value approaches the competitive allocation when it does, is understood for particular families of markets and not in general; the glove market’s rate, a drift of order $n$ against a fluctuation of order $\sqrt n$, is one data point in a picture that has not been completed.

Threats and contributions #

The glove market makes a sharp case out of an abstract distinction. The core is a theory of threats: a split stands if no group can do better alone, and in a market where one side is long, the long side’s members can always be undercut by a coalition that leaves one of them out. A single spare glove is enough to make that threat available against every member of its side, so the scarce side takes everything. The average over orders is a theory of contributions: a player is paid what it adds on average, and in a large market a single spare glove changes what anyone adds only by the faint drift of a random path towards one side of a diagonal.

Both are coherent, both are characterised by axioms that sound reasonable, and on this market they give answers that grow further apart the larger the market becomes. Choosing between them is not a matter of computing more carefully. It is a choice about whether a share should reflect what a player could force or what a player brings — and one glove too many is the smallest market in which the two come apart completely.