There are exactly two ways to reconcile many preferences: aggregate them into one outcome (voting), or sort people among many outcomes (markets/exit). Aggregation is provably defective in ways that scale with the dimensionality of the policy space; sorting escapes those theorems entirely — but increasing returns to coordination keep collapsing sorting back into aggregation. Parties, and legal orders, are both products of that collapse.
Links: Government Formation and Structure (the impossibility floor + scope-of-proportionality work lives there — this page assumes it rather than repeating it), Force Doctrine, Legal Theory, Self-Ownership and Property Rights, Scope Confusion, Technology → Culture → Politics
Specimens (evidence, dated):
The decisive variable is how many independent dimensions the policy space has:
| Policy space | Result |
|---|---|
| 1 continuous dimension, single-peaked preferences | Majority rule works. Black’s median voter theorem yields a genuine Condorcet winner. No cycling. |
| ≥2 dimensions | Chaos. McKelvey–Schofield: majority rule generically has no equilibrium, and an agenda-setter can drive the outcome anywhere. Cycling is the rule, not the exception. |
This reframes the endless proliferation of political models — 1-axis left/right, horseshoe, the 2-axis compass, 4-axis and 8-axis variants. Each new model is an admission that the previous dimensionality was underspecified. Horseshoe theory is best read not as a theory but as a symptom: bending the line is what you do instead of adding a dimension.
If majority rule is only well-behaved in one dimension, then a stable polity must get to one dimension. That is what a party system does: bundling unrelated positions into a platform forcibly projects an n-dimensional preference space onto a single line.
So two-partyism is over-determined. It is not only Duverger’s mechanical effect of first-past-the-post (already treated in government-formation) — it is also the institutional answer to multidimensional chaos. Two independent forces push the same way.
Corollary — unrepresentation is quantization error. The four-quadrant voter who feels no party speaks for him is not the victim of corruption; he is a point in high-dimensional space being rounded to the nearest of two. The information destroyed is the projection residue. This predicts that felt unrepresentation should rise as the salient dimensions multiply, independent of how honest the parties are.
Corollary — defection is self-policing. Because the system has memory, a representative who switches sides gains nothing and loses their coalition. The equilibrium needs no enforcer; the punishment for individual defection is structural.
Every impossibility result in this area — Arrow, McKelvey–Schofield, the discursive dilemma — is a theorem about aggregation functions: mappings from many preference profiles to one collective outcome. This exposes an escape that is categorically different from proposing a better voting rule:
Voting aggregates; markets sort. Aggregation compresses many preferences into one outcome, and the theorems say that compression is always defective. Sorting matches many preferences to many outcomes — nothing is compressed, so nothing is lost.
Sorting does not beat the theorems; it removes their target. If no single collective outcome must be produced, there is no aggregation function left to constrain. This is exit substituting for voice (Hirschman), and Tiebout’s “voting with your feet” carried to the limit by unbundling jurisdiction from territory — the an-cap proposal. It also dissolves quantization error outright, since sorting has no projection step.
The strongest objection to market-provided governance is not the standard public-goods one. It is structural, and it is the same mechanism as Duverger:
The unifying claim of this page: increasing returns to coordination is the single force behind both collapses. It drives voters onto two parties and legal orders onto one system. Duverger’s law and the tendency of law to monopolize are one mechanism wearing two costumes.
That page states the third horn as: a market of competing governance providers dissolves the monopoly problem but can’t handle power asymmetry — the first provider that chooses force over competition conquers the rest. The mechanism above is strictly stronger, because it needs no such provider:
Consolidation does not require anyone to defect to force. Increasing returns to coordination collapse the market for law even under universally good faith. The an-cap horn therefore holds against a population of perfect non-aggressors — removing the “but our providers wouldn’t do that” reply, which only answers the force-based version.
And the aggregation side supplies the other horn’s engine. The McKelvey–Schofield result in §1 is the formal statement of agenda control: where no equilibrium exists, whoever sets the order of votes determines the outcome, so procedural power is substantive power. That is Gödel governance in miniature — the rules for deciding are themselves a decision the rules can’t settle, so the constitution “contains the tools for its own subversion” in a precise, provable sense rather than an analogical one. Empirical note (Chris, 2026-07-20): small-scale democracy experiments reportedly fail predominantly by agenda control — which is the failure mode this predicts should dominate at small N, where no institutional scar tissue obscures it.
Unbundling into independent per-issue votes doesn’t escape either — judgment-aggregation results (the discursive dilemma, List & Pettit; the Ostrogorski paradox) show issue-by-issue majorities can assemble a package a majority rejects.
| Bundled (parties) | Unbundled (issue-by-issue) | |
|---|---|---|
| Coherence | coherent programs | can yield packages a majority rejects |
| Representation | quantization error | fine-grained |
| Stability | stable (1-D projection) | cycling, agenda-control vulnerability |
Not an impossibility theorem in Arrow’s sense, but the same shape: buy coherence or buy representation. Nobody has paid for both.
Where this lands normatively. An-cap does not eliminate the aggregation problem — it minimizes the surface area on which aggregation is required, pushing it down to a thin meta-layer of inter-agency dispute norms, which still must be agreed by something. Min-cap is the honest form of the same insight: the aggregation surface can be made small, but not zero. Note this is the minarchist “the minimum is the minimum” criterion — minarchism as an optimization criterion rather than a fixed quantum of power — applied to aggregation instead of to force. The open weakness is that the criterion is coherent without being self-locating: it does not say where the minimum is, and every state has claimed to sit on it. See self-ownership and property rights — property enforcement is itself coercion, so it must be priced inside the minimum rather than treated as its absence.
The correction that completes the picture, and the one most often missed by people invoking Duverger to dismiss third parties:
Two parties is the equilibrium — but not the same two parties. Duverger constrains how many viable parties there are, not which. Historically the function of a US third party is not to survive as a third party but to displace one of the two (the Republicans killing the Whigs). A third-party bid can therefore be simultaneously doomed as a third party and consequential as a replacement.
This slots exactly into §2’s framing. If a party system is a projection of n-dimensional preference space onto one line, then a realignment is a change of basis — the output stays 1-D (two poles), but which axis gets projected onto rotates. That is precisely what US realignments look like: the salient dimension shifts (slavery in the 1850s, economics in the 1930s, civil rights and culture from the 1960s). The number of parties is invariant; the orientation is not.
Falsifiable prediction: a third-party bid should be assessed not by its odds of winning seats but by whether it rotates the salient axis. A movement that merely adds a third pole on the existing axis is a spoiler and dies; one that makes a new dimension salient can displace an incumbent party outright.