Wednesday’s post answered what selects: a stationary distribution, the wheel turning by no hand in particular. Knowing what the dynamics pick, though, is not the same as knowing whether they picked well. So step off the graph — out of the local view every node is stuck inside — and try to do the one thing the nodes themselves never can: grade the result. Is the equitable convention better than the uniform one, or the other way around? That is a different kind of question from anything the first four posts asked, and the rest of this one is about why the view from outside does not get to answer it cleanly.
Notice what the request smuggles in. To grade the conventions from outside is to import a standard the nodes do not carry — some criterion that pronounces one distribution of \(b – c\) better than another — and the difficulty is that every criterion worth importing either says nothing or says too much. The ones that stay internally consistent rank everything a tie; the ones that discriminate stop being consistent. What follows is a tour of that fork, and there is no third tine.
Put the two conventions on a single dial. Let \(\kappa\) run from \(0\) to \(1\): at \(\kappa = 0\), the equitable split, where the cost-sharing was engineered so that everyone nets \(b – c\) on any graph you can draw; at \(\kappa = 1\), the uniform split, where a node’s take swings with the degrees of its neighbors. Write node \(i\)’s net payoff as a straight line in the dial:
\[ \mathrm{Net}_i(\kappa) \;=\; (b – c) \;+\; \kappa\,\frac{b}{2}\,(\beta_i – 1), \]
where \(\beta_i\) is the sum of inverse neighbor degrees we derived in Post 2. At \(\kappa = 0\) the second term disappears and every node lands on \(b – c\), the identity we leaned on through Posts 2 and 3. Turn the dial up and a node’s fortune starts to ride on \(\beta_i\): a node prefers more uniformity exactly when \(\beta_i > 1\), which happens when its neighbors are on average low-degree — the leaf-huggers, who hand out generous shares because they have so few others to split with. Wire yourself to the hub and \(\beta_i < 1\); you want the equitable split. Wire yourself to leaves and \(\beta_i > 1\); you want uniform. The two conventions are not Pareto-comparable, then, and they never will be: every step toward uniform lifts the leaf-adjacent and sinks the hub-adjacent, and every step back reverses it. No one’s gain arrives without someone’s loss. The average of \(\beta_i\) across the population is exactly \(1\) — it has to be, since the \(\beta_i\) sum to \(N\) — so the dial pivots, on average, around a node sitting at perfect indifference. Where the median node sits is a different question, and a loaded one; I am saving it for Friday.
Now add it all up, and the post earns its subtitle. Sum the net payoffs across the whole population:
\[ \sum_i \mathrm{Net}_i(\kappa) \;=\; N(b – c) \;+\; \kappa\,\frac{b}{2}\Big(\sum_i \beta_i – N\Big). \]
And \(\sum_i \beta_i = N\), exactly, on every graph there is.1 The correction term is zero, not approximately and not on average but identically, which leaves
\[ \sum_i \mathrm{Net}_i(\kappa) \;=\; N(b – c) \quad\text{for every } \kappa. \]
The total surplus does not budge. Uniformity does not grow the pie and does not shrink it; all it ever does is recut it. The pie was the same size all along.
(Ed: five posts to tell me the pie never changes size — I’d ask for a refund, but the aggregate’s invariant, so we’d only end up fighting over who hands it back.)
This is not a lucky cancellation, and it pays to see why. The equitable convention was engineered as an identity in the first place — the cost split chosen precisely so the books would balance for every node on every graph — and \(\sum_i \beta_i = N\) is the fact that carries that engineering intact across the whole dial, not just at its left end. The aggregate is built, by construction, to be unable to adjudicate. A grader who reaches for the total is reaching for a quantity the model was designed to hold fixed, and the design did not slip.
This is worse for the outside grader than it first looks. The utilitarian has nothing to work with: every setting of the dial posts the same total, so the criterion that ranks by aggregate welfare calls them all a tie. Pareto recused itself a paragraph ago, the moment we saw each convention carries its own winners and losers. The two workhorses of welfare economics — add it up, or look for the change that helps someone and harms no one — both go silent in the same breath. What survives them is the question they were built to step around: not how large the pie is, but who walks off with which slice.
Notice the shape of that. Forcing the efficiency question to a clean answer — the total is invariant, full stop — did not retire the problem of evaluation. It relocated it. Every ounce of indeterminacy that might have lived in is it efficient? got pushed, undiminished, into is it fair? The determinacy we won on the aggregate we paid for, in full, on the distribution. Conservation of impossibility once more: a difficulty settled in one place does not vanish, it reappears in another with all of its original weight. We have watched the move three times now — the designer dissolved in Post 3 and the steering came back in the dynamics; the dynamics settled in Post 4 and the question of whether they settled well came back here; and now efficiency answers itself, so fairness inherits the entire bill.
And the place it reappears has a guard posted. Ranking distributions — declaring this division of \(b – c\) better than that one — is the precise business that social choice spent the twentieth century proving cannot be done cleanly. Arrow is the general fact underneath it all: no rule turns a population’s individual rankings into a coherent social one while honoring a short list of conditions nobody wants to give up.2 Sen ground that to a point that cuts here in particular. His impossibility of a Paretian liberal3 shows that you cannot grant individuals even a sliver of local decisiveness — a small private sphere each one governs — and also insist the social verdict respect unanimous preference, without the whole construction cycling. Our nodes come pre-equipped with that sliver: each one acts on its own neighborhood and nothing past it, decisive over its local corner by the very design of the dynamics. Demand a global, Pareto-respecting grade laid over all those local spheres and you are standing exactly where Sen stood, watching the ranking chase its own tail. The outside view wants both things at once, local autonomy and a clean global verdict, and Sen’s theorem is the proof that wanting both is incoherent.
The gears are worth watching turn, because the result is easy to wave at and hard to climb out of. Hand two nodes each a single pair of outcomes they personally decide — their private sphere — and let unanimous preference settle the pairs nobody claims. The first node, decisive over its pair, pushes the social ranking one way; the second pushes its own pair another; and the unanimous links running between the pairs close the loop, so the social ordering ends up ranking \(x\) over \(y\) over \(z\) over \(w\) and right back around to \(x\). A cycle, assembled out of nothing but local authority and broad agreement. No one anywhere behaved badly. The conditions simply cannot share a room, and adding a graph underneath them does not give them more space.
You might expect majority rule to ride in here. Let the nodes vote between the conventions and take whatever most of them want. In one dimension that instinct is exactly right, and it is the entire story of Friday’s post. Turn the evaluation loose across many dimensions, though, and it falls apart: McKelvey and Schofield showed that multidimensional majority rule has no resting place and can be walked to any outcome at all by a patient agenda-setter4 — the chaos this blog took its name from. The one thing between us and that chaos is that \(\kappa\) is a single dial. One dimension. Black’s median voter lives there and nowhere wilder, and that is where I will pick this up on Friday — who carries the vote, and why the dynamics from Wednesday refuse to return the same answer.
Look at the conventions from the winners’ side and they read as progress; from the losers’ side, as plunder; from the sum, a wash; from the spread, a standing fight. Every one of those angles is true, and not one of them ranks the thing. Somebody already wrote the definitive song about seeing a thing from every side and coming away understanding it no better for the trouble. With that, I leave you with this.
Notes
1 Swap the order of summation: \(\sum_i \beta_i = \sum_i \sum_{j \in N(i)} 1/k_j\), and each node \(j\)’s share \(1/k_j\) gets counted once for every one of its \(k_j\) neighbors, contributing \(k_j \cdot (1/k_j) = 1\). Sum over all \(j\) and you get \(N\). The books close at the level of the whole graph precisely because what a low-degree node over-contributes to one neighbor it under-contributes elsewhere — the conservation is exact even though no single node’s ledger balances.
2 Arrow, Social Choice and Individual Values (1951). What it means to call a collective choice legitimate when no aggregation rule comes out clean is the question Maggie and I take up at book length in Social Choice and Legitimacy (Cambridge, 2014); her work lives at elizabethmpenn.com.
3 Sen, “The Impossibility of a Paretian Liberal,” Journal of Political Economy 78 (1970): 152–157. The mapping is structural rather than literal — our nodes are not casting ballots over social states — but the obstruction is the same one: minimal local decisiveness laid under a global Pareto requirement has no consistent solution.
4 McKelvey (1976) and Schofield (1978), establishing that majority rule over more than one dimension is generically unstable and can be agenda-driven anywhere — the result behind this blog’s subtitle. I learned to read these theorems, and to respect the full reach of what they rule out, from Norman Schofield.