Capacity, κ, and the market that freezes

Rule 1 — money vests to the opposing book on arrival — is the idea. Rule 2 is what makes it a mechanism instead of a toll booth. This article is about Rule 2: what it fixes, what it costs, and the failure mode that makes it unsuitable for anything but binary markets.

It is the least glamorous part of the design and the part most likely to break a deployment.

Why Rule 1 alone is broken

Distribute each stake pro-rata over the opposing book, with no ceiling on what a book may absorb. Now consider the first position on an empty book. Until a second position joins, it receives 100% of every opposing inflow. Its return on capital diverges as the position shrinks.

The design-phase measurements, preserved in vpm-edge-tests.mjs:

The shipped conformance suite pins the same divergence as P5's 200,001× single-probe comparison.

So the profit-maximizing strategy under Rule 1 alone is to post dust on every new market and toll the organic flow, supplying no liquidity whatsoever. That is the precise opposite of what a cold-start mechanism is for.

Rule 2

Each position grants its own book κ · s_i of matching capacity. A stake is accepted only up to the capacity remaining in every opposing book; the unmatchable remainder is refused at entry and returned in the same transaction.

κ (kappa) is one published constant — a maximum-odds cap, the same object every betting exchange already publishes. There is no bucket, no decay schedule, no ρ, no residual refund. The paper tried the alternative (a metered bootstrap bounty releasing unmatchable flow to later entrants at up to ρ×, decaying with market age) and killed it in Appendix A.2 for three separate reasons — the sharpest being that its decay is indexed to clock time while the risk it prices is informational, so a market that resolves informationally early reinstates most of the free-ride. A capacity rule has no schedule and therefore has no schedule to mis-index.

The placement is the whole trick

This sentence from §4.1 is worth reading twice:

Capacity constrains acceptance at the book level, never the distribution, which stays pure pro-rata.

Why it matters:

Pro-rata distribution makes return scale-invariant. A dust position earns exactly the multiple that any capital entering at that moment earns — never more. So there is nothing for dust to dominate. The problem was never the pro-rata split; it was that an empty book could absorb unlimited flow.

The common multiple is then bounded. While a book's principal is constant, accepted opposing inflow cannot exceed remaining capacity κ·P − V, and each new unit of same-side principal adds at most κ more, priced at the enlarged book. Integrating gives a winner's multiple of at most 1 + κ(1 + ln(P_T/P_τ)). Note that this is not a flat 1 + κ except on a static book — it grows with the book's own growth. Verified across 14,520 positions in adversarial random markets, with zero violations, and an ε-position probing a seeded market returns 9.0–9.7× at κ = 9 across stakes from 1 to 10,000 units.

Constraining acceptance keeps the O(1) form exact. The obvious alternative fix — cap each position at κ·s_i — also closes the dust attack, and it is the wrong one. It binds in 87% of test settlements at κ = 3, and whenever it binds the allocation stops being pro-rata, which breaks the constant-time accumulator in §6 and with it the property that a payout is a function of two scalars. That failure is pinned as suite case A4.

What it costs in a binary market

The favoured side receives partial fills once the underdog book is saturated. If you try to put $500 on the obvious side of a thin market, you may get $400 of it. That is an honest liquidity signal, and it is the correct answer: there was only ever $400 of honest exposure available at those odds.

The blocking economics deserve stating precisely, because they are asymmetric.

It is not free to weaponize if you do not want the position. To exhaust the headroom another buyer needs, you must stake their side yourself, at full risk. Conformance case P10 pins the squatter eating the loss when the blocked side loses.

It is free at the margin if you wanted the exposure anyway. By scale-invariance the blocking capital earns the same multiple whether or not it crowds a rival out (case P10b), and seizing the remaining headroom converts shared future vesting into an exclusive claim on it.

So under a binding κ, entry becomes a race for headroom among same-side traders. The harm is redistributive between them, and informational — the blocked rival's view never enters the pool. The block-vintage rationing rule removes the within-block race; per-account acceptance caps are the venue-level mitigation for the rest, at the cost of sybil pressure.

In practice, at sensible κ, none of this touches ordinary flow. Rule 2 refuses 0.10% ± 0.02 of gross stake at κ = 9. At κ = 3 it refuses 10.1%, with the decile pattern unchanged. From κ = 12 up, measured refusal is zero to two decimals.

What it costs in an n-way market — the part that matters

Here the constraint couples across branches, and the consequence is not a degradation. It is a kill switch.

Acceptance takes the minimum headroom over every opposing book. So entry on any outcome requires positive headroom in each opposing book, which means the thinnest book gates entry on every outcome that opposes it.

Two measured consequences:

Liveness. A three-outcome market with one unloved branch refuses 97.5% of gross volume at κ = 9 (the pinned skewed-book construction, case P9b). Note how far that is from the balanced n = 3 arm's 1.03% refusal — the balanced number is not representative, and the paper says so explicitly.

Griefing. A $450 stake on a side outcome blocked a $2,000 informed entry entirely (case P9). The same entry clears at κ unbounded.

The absorbing freeze

The sharper statement, and the reason this section exists as its own article. From §4.3, pinned as conformance case P9c:

A symmetric seed S leaves every book exactly S(κ − n + 1) of headroom. One stake of that size on a single outcome consumes the headroom of the n − 1 books opposing it. Thereafter every outcome has a saturated book among its own opponents, so:

No entry on any outcome at any size is ever accepted again.

Capacity grows only with a book's principal, and principal can no longer grow, because the stake that would replenish a starved book is itself refused. The market is not throttled. It is dead, and it cannot be revived from inside the mechanism.

At the paper's own reference parameters — κ = 9, n = 3, a $50 seed per leg — the trigger is $350, and the attacker's own accepted stake is capped at exactly that. Worse, the freeze is reachable at any point in a market's life, not just off the seed: after six ordinary entries the same construction costs $770 and is equally permanent.

n = 2 is the sole exception, and it self-heals — because the single opposing book is precisely the one a recovering entry needs. That asymmetry, not a difference of degree, is what the paper's conclusion rests on.

The rule for venues

Finite κ is a binary-market instrument.

Markets with three or more outcomes — and ladders with far-out-of-the-money bands especially, since those are exactly the skewed books P9b describes — should run κ large or unbounded.

The obvious worry is that this removes the guardrail. It doesn't, and this is the part worth internalizing: the mechanism's soundness does not depend on the cap. What holds it up without κ is the pair that Rule 2 arrived with and that stays:

1. The mandatory all-outcome seed, so no empty book can ever exist.
2. Pro-rata scale-invariance, so dust earns what any same-moment capital earns and P5's dominance analysis needs no cap at all.

With κ unbounded, thin-book multiples are simply long odds — exactly as in a classic parimutuel. What remains bounded is what actually matters: nobody's accrued claim, and nobody's dominance.

Choosing κ

κ is venue policy, published like a fee table before the market opens, not a mechanism identity. A short decision table:

One arithmetic constraint that catches implementers: κ ≥ 1 is a domain requirement, not a suggestion. Below it, no symmetric seed passes the clamp at all and creation voids.

The repair that does not exist yet

There is an obvious fix for the freeze, and the paper states it and then declines to design it (Appendix B.5): a matched all-outcome top-up, obeying the vintage-0 rule with legs as counterparties to each other, restores c(κ − n + 1) of headroom per book while consuming (n − 1)c. It strictly unfreezes.

Three questions block it, and they are real:

• Does a second matched vintage preserve P6? The creation floor's proof relies on the creator owning the entire market at a unique reserved vintage.
• Does it preserve P3 for holders who entered between the two matched vintages?
• Who may call it? A creator-only top-up is a dilution lever over standing positions. A permissionless one is a new griefing surface, since the same construction that freezes a market is cheap for whoever wants it frozen.

The shipped settler admits no second matched vintage. Until that analysis exists, large κ is the answer for n-way markets, and it is a complete answer — it just is not the elegant one.

Next: Four ways to make a market — where this sits among the alternatives.