A block in an ocean of small holders

A share of the votes is not a share of the power counted, member by member, how often each voter in a small assembly is the one whose arrival turns a losing coalition into a winning one, averaged over every order in which the members could arrive. A vote worth nothing until it was outnumbered did the same for the Council of the European Communities and found Luxembourg’s single vote powerless. Both were assemblies of a handful of members with comparable weights. Most real decisions are taken in bodies of another shape: a company with a few large shareholders and thousands of small ones, a parliament of a few parties and many independents, a market with a few large traders and a crowd.

For those, the average over orders has to be taken over an astronomical number of orders — too many to list, as the sampling essay found at twenty players. But the crowd’s very size turns out to make the problem easier rather than harder. John Milnor and Lloyd Shapley called the dispersed holders an ocean in 1961 and found that, in the limit of infinitely many infinitesimal holders, the average over orders becomes a simple integral, and the power of each large block has a formula a reader can evaluate in the head.

Thirty per cent buys forty-three #

The figure takes a company in which a motion passes when more than half the shares are in favour, one block holds a share $a$, and the rest is divided equally among 200 small holders. For each $a$ it computes the block’s Shapley–Shubik index exactly: the chance, over a uniformly random order of arrival of all 201 holders, that the block is the one whose arrival first pushes the shares present past a half. The computation does not list orders. Whether the block is decisive depends only on how many small holders arrived before it, and the chance that exactly $s$ of the 200 come first, when the block’s own position is uniformly random, is a ratio of binomial coefficients; summing over $s$ gives the answer in a few hundred terms.

The dots climb above the diagonal from the start. A block of 10% holds 11.4% of the power; a block of 20%, about 25%; a block of 30%, 42.3% with two hundred small holders and $3/7 = 42.9\%$ in the limit. By a third of the shares it holds half the power, and at a half of the shares it holds all of it, as it must: a block with half the votes decides every motion alone once it is joined by anyone. In between, the curve has no straight part. Power concentrated in a block is worth more than the same votes dispersed, and the excess grows with the block’s size until it saturates.

The shape of the curve answers an old question in the study of companies. Adolf Berle and Gardiner Means argued in 1932 that in a corporation with widely dispersed shares, a holder of a fifth or a quarter could exercise working control, because the dispersed holders cannot coordinate — and company law in many countries has since treated thresholds of that size as control for the purposes of takeovers and disclosure. The oceanic curve gives the structural version of the argument without assuming anything about coordination: a quarter of the shares is a third of the power before anyone organises anything, simply because the dispersed shares arrive in a smooth stream that the block can always top up. Whether real dispersed holders behave like a smooth stream is a question about turnout and organisation that the average over orders does not answer; what it shows is that the advantage of concentration is built into the arithmetic of a quota, and does not depend on the small holders being apathetic.

The moments at which a block decides #

The limit has a picture that makes the formula obvious. Instead of a random order, give every holder an independent random arrival time between 0 and 1; the order of the times is a uniformly random order, so nothing changes. With a great many small holders, the shares of the ocean present by time $t$ are, by the law of large numbers, almost exactly the fraction $t$ of the ocean’s total: the ocean trickles in at a steady rate. The block, arriving at its own random time $\tau$, adds its whole share at once.

The block is decisive exactly when, at its arrival, the ocean present is still below the quota but the ocean plus the block is above it. With the ocean holding $1 - a$ and the quota $q$, that means $(1 - a)\tau \le q < (1 - a)\tau + a$, a window of arrival times

$$\frac{q - a}{1 - a} \;<\; \tau \;\le\; \frac{q}{1 - a},$$

clipped to $[0, 1]$. Since $\tau$ is uniform, the block’s power is the window’s width. For $a = 0.3$ and $q = \tfrac12$ the window runs from $0.286$ to $0.714$, a width of $3/7$. Each small holder is decisive only in the instants just before the block’s window and just after it, and the ocean as a whole shares the remaining $4/7$.

Two consequences can be read straight off the picture. The width of the window is $a/(1 - a)$ whenever both ends are inside $[0, 1]$ — that is, whenever the block is smaller than both the quota and its complement — and the quota does not appear in it at all. And as $a$ approaches a half under majority rule, the window’s upper end passes 1 and its lower end passes 0, so the block is decisive at every arrival time.

A second block #

With two blocks the window becomes a sum over whether the other block has arrived yet, each case weighted by the chance $t$ that it has, and the figure computes it both ways: exactly with 300 small holders, and as the oceanic integral. The answers agree to within a hundredth everywhere.

The second block’s power comes mostly at the first block’s expense, not the ocean’s. A rival of 10% takes 8% of the power and costs the first block only one point, but by the time the rival reaches 20% the first block has fallen from 43% to 36%, and at two blocks of 30% each holds exactly a quarter — less than its share of the votes — while the ocean’s 40% of the shares carries half the power. Push the rival to 40% and the original block holds 11%. A block’s power depends less on its own size than on what it is up against: the same 30% is a controlling stake against a dispersed ocean and a minor one against a larger rival. Corporate lawyers know this as the difference between a block and a control block; the average over orders puts a number on it, and the number moves with every change in the rest of the register.

The ocean’s share is not monotone either. It falls from 57% to 48% as the rival grows to 20% — the rival takes some pivotal moments from it — and rises back to half when the two blocks are equal, because two equal blocks of 30% need the ocean to break their tie, which is the same effect that gave Luxembourg its first power when the Council grew.

How large an ocean is an ocean #

The limit is approached quickly but not smoothly. With a single other holder of 70%, the block has no power: the other holder passes everything alone. With two holders of 35% each, any two of the three make a majority, and all three are symmetric, with a third of the power each. With three holders of $23\tfrac13\%$, the block plus any one of them is a majority, while the three small holders need all three — the block is decisive in half the orders. The values then swing as $N$ changes which combinations of small holders can tip the balance, and from about twenty holders on they stay within two points of $3/7$, reaching 42.6% at four hundred.

That is the practical content of the limit. A company with a few dozen dispersed holders behind a large block is already, for purposes of power, an ocean, and the formula $a/(1 - a)$ describes it. The swings at small $N$ are the arithmetic of weights and quotas that the earlier essays explored — accidents of which subsets can reach the quota — and the ocean smooths them away because no single small holder’s weight matters.

Two power indices on one diagonal #

The arrival-time picture is not only a device for oceans. Guillermo Owen showed in 1972 that it gives the average over orders for any game, exactly. Fix a member $i$ and a moment $t$; if every other member has arrived independently with chance $t$, the chance that $i$ is decisive is a polynomial in $t$, the slope of the game’s multilinear extension — the unique function of $n$ variables, linear in each separately, that agrees with the game on the corners of the cube. Averaging over the uniformly random arrival time of $i$ itself gives

$$\varphi_i = \int_0^1 \frac{\partial f}{\partial x_i}(t, t, \ldots, t)\, dt,$$

the integral of that slope along the cube’s main diagonal. The figure draws the five slopes for an assembly with weights 5, 4, 3, 2 and 1 and quota 8. Each curve starts at nought, since nobody is decisive in an empty room, rises, and returns to nought, since nobody is decisive in a full one. The areas under them are 0.400, 0.233, 0.233, 0.067 and 0.067 — exactly the Shapley–Shubik indices, checked against all 120 orders of arrival. Members of weights 4 and 3 have identical curves, and so do 2 and 1: in this assembly the weights differ and the powers do not, the effect the earlier essay found with weights 4, 4 and 1.

The heights of the same curves at the single moment $t = \tfrac12$ — every other member present or absent with even chances, every coalition equally likely — are John Banzhaf’s index of 1965, which counts the coalitions in which a member is a swing. Scaled to add to one they are 0.385, 0.231, 0.231, 0.077 and 0.077. The two classic indices are the same curves, averaged in two ways: Shapley and Shubik integrate along the whole diagonal, Banzhaf reads the middle. Where the curves are symmetric about the middle and nearly flat-topped the two agree; where a member’s decisiveness is concentrated early or late they differ, and the figure shows the large member’s curve peaking a little after the middle, which is why its Shapley index exceeds its Banzhaf share.

The same diagonal prices costs and explains predictions #

Owen’s integral has a life outside voting. When the players are not people but quantities — the amounts of several services a customer uses, or the inputs a model is given — the “game” is a smooth function, and the average over orders of arrival of infinitesimal units becomes the integral of each partial derivative along the straight line from nothing to the actual amounts. That is the Aumann–Shapley price, from Robert Aumann and Lloyd Shapley’s book on non-atomic games of 1974: each unit of a service is charged the average of its marginal cost as the whole bundle is scaled up from zero. Louis Billera, David Heath and Joseph Raanan used it in 1978 to set the internal charges for Cornell University’s telephone system, where the cost of a shared network had to be divided among departments using it in different proportions — the same problem sharing a cost that is not the sum of its parts solved by the average over orders for a handful of users.

It reappeared in 2017 in machine learning as integrated gradients, a method for saying how much each input of a trained model contributed to one prediction: integrate the model’s gradient along the straight line from a reference input to the actual one, and assign each input the area under its own slope. It is the Aumann–Shapley price under another name, and it inherits the axioms — efficiency, symmetry, the treatment of a missing input — that what a missing input is worth found the discrete average over orders to satisfy, along with the same difficulty: the answer depends on the reference point from which the line is drawn, which is a modelling decision and not a fact about the model. A block in an ocean, a department on a telephone network and a pixel in an image are charged by the same integral along the same diagonal.

The quota matters only for large blocks #

The window’s width, $a/(1 - a)$, does not involve the quota, and the figure shows what that means. Under a simple majority, a two-thirds rule and a three-quarters rule, a block of 10% holds 11.1% of the power, a block of 25% holds 33.3%, a block of 30% holds 42.9% — identical in all three. Only when the block grows large enough to pass a motion alone, or to stop one alone, do the rules separate. Under majority rule a block of a half decides everything; under a three-quarters rule a block of a half holds exactly half the power, and needs 75% before it decides everything.

That is the oceanic form of a familiar fact about supermajorities: they protect the dispersed against large blocks only for blocks near the blocking size, and not at all for small ones. A rule that requires two thirds instead of a half changes nothing for a holder of a fifth of the shares facing an ocean, and a great deal for a holder of a third, who becomes a blocking minority. It is one more case in which the rules and the weights together, and not either alone, decide who has power.

What the computations show and assume #

The finite values are exact: each is a sum of binomial ratios over the number of small holders arriving first, and every total is checked to divide the whole power between the holders. The oceanic values are integrals over the arrival time, evaluated numerically to four places, and they agree with 200 and 300 small holders to within two hundredths, which is the size of the swings the finite-ocean figure shows. The assembly of five is checked against all 120 orders.

The model is the one every earlier essay here on the Shapley value has used: power as the chance of being decisive when every order of arrival is equally likely. Real shareholders do not arrive in random order; they vote in blocs, abstain, and follow advisers. The average over orders is a measure of structural power — what the weights and the quota allow — and not a prediction of how any vote goes. The ocean assumption is also an idealisation: real small holders are not equal and do not all vote, and turnout among them is itself a large-block phenomenon, since an organised group of small holders behaves like a block.

Still open: values for oceans that are not oceans #

Milnor and Shapley’s limit, and its general form in Robert Aumann and Shapley’s theory of non-atomic games of 1974, describe games in which the small players are truly negligible. Many games of interest have small players who are not: a market with a few hundred traders of different sizes, a parliament with dozens of small parties. How fast the finite values converge to the oceanic ones, and with what error, is known for weighted majority games of particular shapes and not in general; the swings in the finite-ocean figure, which last until about twenty holders here, can last much longer when the small holders’ weights are unequal or the quota sits on an arithmetic edge. And for games whose value function is not smooth in the ocean’s share — quotas that are exactly met, thresholds that jump — Aumann and Shapley’s diagonal formula needs to be repaired, and the question of which repair is right has more than one answer in the literature.

The same average, run along a diagonal #

A block of shares among dispersed holders has more power than its votes: 30% of the votes gives 3/7 of the power, a third gives half, and a half gives all — because with the ocean trickling in at a steady rate, the block decides every vote in which it arrives within a window of width a/(1 − a). A second block takes power mostly from the first, two blocks of 30% each hold a quarter, the quota is irrelevant until a block can pass or stop motions alone, and the limit is reached in practice by a few dozen small holders. Owen’s diagonal makes the same picture exact for every game: the Shapley value is the area under a member’s decisiveness along the diagonal of the cube, and the Banzhaf index is its height at the centre.