Thursday’s post ended on a one-dimensional dial and a promise. The promise was that a single dimension is exactly what majority rule needs in order to behave. Turn an evaluation loose across many dimensions and it cycles — McKelvey and Schofield, the chaos in this blog’s masthead1 — but compress the whole question of equitable-versus-uniform onto the lone parameter \(\kappa\), and the chaos has nowhere to start. One dimension is the difference between a vote that means something and a vote that a clever agenda-setter can walk anywhere it likes. So let us hold the dial steady and, six posts in, finally take the vote.
Each node’s net payoff is affine in \(\kappa\): \(\mathrm{Net}_i(\kappa) = (b – c) + \kappa\,\frac{b}{2}(\beta_i – 1)\), a straight line whose slope is \(\frac{b}{2}(\beta_i – 1)\). A line has no interior peak, so every node’s most-preferred setting sits at one end of the dial. A node with \(\beta_i > 1\) climbs toward \(\kappa = 1\) and wants the uniform split; a node with \(\beta_i < 1\) slides toward \(\kappa = 0\) and wants the equitable one; a node resting exactly at \(\beta_i = 1\) is flat, indifferent across the entire range. Preferences this plain are single-peaked by default, which is the one hypothesis Black’s median voter theorem asks of us.2 On a single dimension with single-peaked preferences the median voter’s ideal point is the Condorcet winner — it beats every alternative head to head — and because every ideal here is pinned to an endpoint, the median ideal is whichever endpoint holds the majority. The whole vote collapses onto one question: do more than half the nodes have \(\beta_i < 1\)?
Thursday pinned the mean. The \(\beta_i\) average to exactly \(1\), an identity true on every graph, since they sum to \(N\) and there are \(N\) of them. A mean of one might tempt you to picture the population splitting evenly around indifference, half above and half below. It does not split that way, and the reason is the shape every network worth studying happens to have.
Take the star, the cleanest case and the one Wednesday already drew. A hub of degree \(D\) sits at the center, and \(D\) leaves hang off it, each touching nothing but the hub. A leaf’s only neighbor is that hub, of degree \(D\), so the leaf’s \(\beta\) is \(1/D\) — small, well under one. The hub’s neighbors are all leaves, each of degree one, so the hub’s \(\beta\) is \(D\) — large, well over one. The arithmetic average of those is one, as promised, but the median is \(1/D\).3 The lone hub sits far out in the right tail while \(D\) leaves pile up near zero, and the median voter is a leaf. The leaves want equitable, the hub wants uniform, and the leaves outnumber the hub \(D\) to one. The majority votes equitable, and the margin is a landslide.
The star is extreme, but the conclusion does not lean on the extremity. Any network whose degrees are right-skewed — a few richly connected nodes over a long tail of sparse ones, which is to say nearly every social, biological, or infrastructural network ever measured — pushes the median \(\beta\) below the mean by the identical logic. The hubs inflate the average from the right; the many low-degree nodes hold the median down on the left. Mean one, median under one, majority equitable. The vote knows what it wants, and what it wants is the convention that pays the periphery.
And the wheel does not take a poll. Wednesday’s dynamics do not count nodes; they weight influence, and influence on a star is the hub and very nearly nothing else. A convention sitting on the hub is copied by every leaf that looks up to it, so it sweeps; a convention sitting on a leaf is seen by no one and dies. That much was Post 4. Here is the part that turns the screw: imitation copies the successful, not the good-for-me, and the hub is spectacularly successful under the uniform split — its payoff climbs with \(D\) while every leaf’s payoff sinks. So the leaves, watching the most conspicuous node in their small world prosper, copy its convention straight into their own ledgers, adopting uniform even though uniform pays them less than the equitable split they would have voted for.4 The dynamics install the convention the majority rejects, and they do it through the majority’s own hands.
The node that loses the vote \(D\)-to-one wins the outcome outright. Counting heads, the hub is a rounding error; weighting edges, the hub is the whole graph.
Set the two rules side by side, because the gap between them is the thing this whole series was built to show. One profile of preferences — every node’s affine taste over the dial — feeds two different aggregations. Majority rule counts: one node, one vote, and the network is invisible to it, because a ballot does not care who your neighbors are. The imitation dynamics weight: who gets copied, who amplifies, and there the network is the only thing that matters. Same preferences, two channels, two clean and contradictory verdicts. The lesson is not that aggregation is impossible — Black handed us a perfectly clean answer, a Condorcet winner with no cycle anywhere in sight. The lesson is that the answer was never a function of the preferences alone. It is a function of the preferences and the channel you run them through, and the network that disappears under counting becomes decisive under copying. Ask the population to vote and it chooses equitable; let the population imitate and it chooses uniform; and nothing in the preferences, nothing whatsoever, tells you which of those the population is “really” choosing.
It helps to name the two operations as operations. Majority rule is a map from the profile of preferences to an outcome that ignores the graph entirely — feed it the same tastes on a star, a ring, or a random tangle and it returns the same winner, because it sees only the multiset of ideal points. The imitation dynamics are a map from that same profile to an outcome that reads almost nothing but the graph — hold the tastes fixed and rewire the network, and the selected convention moves with the wiring. A social choice, then, is not a function of what a population wants. It is a function of what a population wants and how that wanting is permitted to propagate, and the second argument does at least as much work as the first. Treating the two operators as interchangeable — reading a population’s “choice” off the outcome without first asking which channel produced it — is the category error this whole arc has been circling.
The whole divergence is the work of the hubs, which you can confirm by taking them away. On a regular graph, where every node carries the same degree \(k\), each \(\beta_i\) is \(k\) copies of \(1/k\), which is one, for everybody. Every node indifferent, median equal to mean equal to one, the vote a shrug and the dynamics a wash. Counting and copying agree because there is nothing left for them to disagree about. Heterogeneity is what pries the two verdicts apart, and the wider the degree distribution spreads, the wider the daylight between what the network votes for and what it imitates its way into. The hub is not a complication sitting on top of the result. The hub is the result.
There is a practical edge on this for anyone who studies a settled population from outside. Walk up to a network that has fixed on the uniform convention, and the temptation is to conclude that its members prefer uniform — revealed preference, the outcome read back as the wants that produced it. On a heterogeneous graph that inference is unsound. The convention you observe may be the one the majority voted against and imitated its way into, the hub’s preference wearing the whole population’s clothes. What presents as consent can be the residue of influence, and from the equilibrium alone the two are indistinguishable. The convention sits there on the table; how it got there is nowhere written on its face.
Which is the moment to admit that the question this series opened with was malformed. “Which convention is better” presumes a single better, a fact about the network waiting to be read off it. There is no such fact. Post 5 showed you cannot grade the conventions from outside, because efficiency goes silent and fairness has Sen standing in the doorway. This post shows you cannot pull a single answer out of the inside either, because inside there are two populations wearing the same nodes — the one that votes and the one that copies — and they want opposite things. The impossibility we have been conserving since Post 3 does not dissolve at the end of the arc. It terminates, instead, in the recognition that “better” was never a one-place predicate — only ever better-by-counting or better-by-copying, with the network setting how far apart those two are allowed to stand.5 The hub, recall from the first post, is the node everyone looks to, the one not everybody can be. That is exactly why the dynamics obey it over the crowd. The trait that makes a node worth emulating is the same trait that lets its convention overrule the vote, and the crowd installs the hub’s preference by the plain act of admiring it.
The median voter is stuck in the middle — that is very nearly the whole content of Black’s theorem — and the wheel that overrules her was never, at any point, going to take a poll. There is a band whose name I have been holding back for exactly this post. With that, I leave you with this.
Notes
1 McKelvey (1976) and Schofield (1978), which together establish that majority rule over more than one dimension is generically unstable and can be agenda-driven to any outcome at all. The one-dimensional \(\kappa\)-line is precisely the special case those theorems exempt — the place where the chaos cannot get a foothold. I learned to read these results, and to respect the full reach of what they rule out, from both Richard and Norman.
2 Black, “On the Rationale of Group Decision-making,” Journal of Political Economy 56 (1948): 23–34. An affine utility is monotone, hence single-peaked with its peak at an endpoint, so Black’s theorem applies and the median ideal point is the Condorcet winner. With every ideal confined to \(\{0, 1\}\), the median ideal is simply the endpoint that commands the majority.
3 For the star \(K_{1,D}\): each of the \(D\) leaves has as its sole neighbor the degree-\(D\) hub, so \(\beta_{\text{leaf}} = 1/D\); the hub’s \(D\) neighbors are all degree-one leaves, so \(\beta_{\text{hub}} = D\). The mean is \(\frac{D \cdot (1/D) + D}{D + 1} = 1\), and for \(D \ge 2\) the median of the \(D + 1\) values is \(1/D\).
4 Payoff-biased imitation and the replicator dynamics are payoff-monotone — strategies that earn more spread — which is a different thing from best response. A node copying its most successful neighbor can adopt a convention that lowers its own payoff, the wedge Schlag’s proportional-imitation results make precise (Schlag 1998, JET). The hub prospers under uniform; the leaves copy the prosperity, not the arithmetic.
5 The two populations wearing the same nodes — the voters and the imitators — are the seam Maggie and I are working in a companion paper, where a node imitates not a neighbor’s convention but its neighbor’s perceived centrality, and the channel itself becomes the object of strategy. Her work is at elizabethmpenn.com.