What a stake is worth when it arrives

Every market structure has a pricing content — the thing you are actually buying when you put money in. In an order book it is a price. In an AMM it is a point on a curve. In a classic parimutuel it is a guess at the closing pool ratio, which is the whole problem with a classic parimutuel: you commit at a price nobody can observe, because it does not exist yet.

Here the pricing content is one formula, and it is the mechanism's central result.

The formula

A stake s on side o accrues the path integral of opposing inflow, weighted by your side's principal as it stands:

V  =  s · ∫ dΠ_opp / P_o

When the pool's composition q — your side's share of the pool — is constant over the interval, that closes:

V  =  s · ∫ (1−q) dΠ / (qΠ)  =  s · ((1−q)/q) · ln(Π_T / Π_t)

Write L = ln(Π_T / Π_t) for the pool's remaining log growth after you enter, and

y  =  ((1−q)/q) · L

for the vesting yield. Break-even is p(1 + y) = 1, which rearranges into the one line an entrant actually needs:

An entry is profitable iff p/(1−p) > [q/(1−q)] / L.

Your odds-belief has to beat the pool's current odds, discounted by how much the pool is still going to grow. That is the whole entry rule.

The published script vpm-yield.mjs holds composition exactly constant and measures pure discretization error: 0.005%–0.02% relative at 10,000 steps, across q ∈ [0.3, 0.9] and L ∈ [0.5, 2], converging as O(1/steps). The closed form is right; what it does not capture is the subject of the second half of this article.

Three consequences

You are priced at the ratio you can see. Under classic rules break-even is p > q_T — you are priced at the closing ratio, unknowable when you act. Here you are priced at the ratio prevailing when you enter, discounted by remaining growth. This is what the paper means by entry-time price integrity, stated precisely: integrity with respect to pool composition at entry. It is not integrity with respect to the conditional law of future flow, which is the next section's subject and which the paper is careful not to claim.

L = 1 is the natural scale. If the pool will grow exactly e-fold after you, the whole condition collapses to

p > q

Enter if and only if you are more bullish than the pool is right now. That is textbook aggregation behaviour, and it falls out of the rule rather than being designed into it. It is also the cleanest intuition available for what the mechanism is doing: an e-fold of growth is the unit in which "beating the current pool" is exactly the right test.

L > 1 opens a two-sided band — and this one is uncomfortable. Both sides are simultaneously positive-EV whenever

p/(1−p)  ∈  ( [q/(1−q)] / L ,  L · [q/(1−q)] )

which is non-empty exactly when L > 1. Early in a fast-growing market, the two implied prices do not sum to one.

Say it plainly, because the paper does: the primary pool ratio is not a probability. It is a pool composition. Anyone reading a VPM market's displayed ratio as "the market thinks 68%" is reading the wrong object. Where a probability actually comes from is §8 and its own article.

What the formula does not price

Two corrections separate the closed form from a usable entry rule. Both are measured in the published suite rather than gestured at, and both cut the same direction: the naive number is too optimistic.

Composition drift

q moves after you enter. When it does, your yield is the path integral, not the closed form.

Under the study's crowd the closed form is roughly unbiased in aggregate — mean signed error of −0.03 in yield units — but it is poor pointwise. The median absolute gap between realized and predicted yield is 55% of the prediction, with a mass point at 100% from positions whose side simply receives no further opposing flow at all.

The right reading: the formula is a pricing model, not an oracle. An agent's edge in using it is exactly as good as its forecast of future flow, and nothing more.

The winner's curse

This is the correction that changes decisions, and it is the one most likely to be skipped by someone implementing the entry rule from the formula alone.

The break-even condition treats your yield and your win event as independent. They are negatively correlated by construction:

1. Vesting comes from opposing inflow.
2. Opposing inflow is disproportionately informed.
3. Informed opposing flow concentrates in exactly the histories where your side loses.

You are paid contingent on winning, so the decision-relevant quantity is not E[y] but E[y | win]. In the paper's crowd model:

Roughly half. For a first-tercile entrant, the naive break-even belief of 0.337 corrects to 0.473 — an understatement of 13.5 percentage points. If you priced off the naive number you would take a large class of entries you should have refused.

The correction shrinks as opposing flow becomes noise, and vanishes entirely for pure-noise flow. So it is not a universal constant; it is a function of how informed the other side is. The paper's instruction to operators is to treat it as a floor on required edge, not a refinement.

And the L = 1 corollary survives only as the zero-drift, uninformed-flow benchmark. That is what "textbook" was doing in that sentence.

Why this matters more in thin markets

The formula needs L̂, an estimate of remaining growth. The obvious estimator is L̂ = ln(T/t) — assume the pool grows in proportion to remaining time.

In thick markets that is roughly fine. In thin ones it is a disaster, and the paper measures it in the regime the mechanism is for. In the sparse long-tail arm (arrival rate 0.03, roughly 27 stakes a market), agents using the naive estimate realize −11.5% per entry post-fee, while the same beliefs under classic rules earn +11.2%.

That is not a small miscalibration. The naive estimator wildly overstates growth precisely where flow is thin, and the winner's-curse correction is not optional there.

The prescription from §10: form L̂ from realized per-class growth curves, never from promotional volume — which, as the wash-trading analysis in P5 shows, a book-dominant party can manufacture at near-zero mechanism cost.

There is one piece of good news on the estimator. When §13.2 iterates L̂ to self-consistency — replacing it with the realized growth curve of the previous round and repeating — it converges in four iterations rather than unravelling. Volume contracts about 20%, the first-third share rises to 50.4%, informed PnL improves from +1.3% to +4.6%, and the fixed-point growth curve sits well below the naive one (at t = 0.1T: 1.61 against 2.30). That is simulation evidence, not a theorem, and the paper labels it as such.

The favourite-longshot side effect

Because y ∝ (1−q)/q, the mechanism pays more per winning dollar for backing the minority side. That is arithmetic, not a policy choice, and it has a consequence the paper takes seriously in §14.

The payment is information-blind. In any crowd where contrarians skew noisy, the minority premium pays noise more per winning dollar than it pays information. The behavioural study measures exactly this: early noise winners land at 2.22×, early informed winners at 2.10×.

What the corrected calibration shows, though, is not a realized-frequency tilt. At 0.95T, in the 0.2–0.3 pool-implied bin, outcomes realize at 0.02 under classic rules and 0.03 here (binomial SE ±0.002–0.003); in the 0.7–0.8 bin, 0.99 against 0.97. The visible effect is compression — the displayed ratio dragged toward the seed's uninformative 50/50 prior as rational late flow stays out, with thinner tail bins. The measured price anomaly is the widening |q − p| gap, not a longshot subsidy.

Using it

If you are writing an entry rule, the honest version is:

1. Estimate L̂ from realized growth curves for this market class, not from ln(T/t) and not from observed early volume.
2. Compute y = ((1−q)/q) · L̂.
3. Halve it, or apply the class-specific E[y|win]/E[y] ratio if you have measured one.
4. Require p̂ · (1 + y_corrected) > 1 + fee + edge.
5. Accept that step 1 is where your entire edge lives, and that the mechanism gives you no help with it.

Step 5 is the honest summary of the whole formula. The mechanism prices you off something knowable — that is the improvement over a classic pool, and it is real. It does not price you off something predictable.

Next: Capacity, κ, and the market that freezes — the second rule, and the failure it creates.