No market average when one prize has no independent price

Inputs

Odds table

Working

1. These figures are assumed, not a recorded product. The odds still sum to 1, so the model can run.
2. The seller's own arithmetic is still reportable: 0.01 × $2,500 + 0.09 × $200 + 0.90 × $60 = $97. That is the catalogue average, and it is reported as such.
3. The top prize has no price recorded outside the seller. Substituting the $2,500 catalogue figure for a missing market price would quietly reintroduce the number the independent column exists to avoid.
4. So the market expected value is not published, and neither is the realisable figure that would have been built from it. Ninety-nine percent of the probability mass is independently priced; the missing 1% is the prize that would have dominated the mean.
5. A record with odds but no independent prices, or prices but no odds, is refused rather than completed. The same rule applies to a single unpriced outcome carrying probability.

Result

The formula behind it

• M = Σ pᵢ (disclosed probability mass; the model refuses unless M = 1)

• EV_declared = d × Σ (pᵢ × vᵢ)

• EV_market = d × Σ (pᵢ × mᵢ)

• realisableᵢ = max(0, mᵢ × b − s ÷ d)

• EV_liquid = d × Σ (pᵢ × realisableᵢ)

• Implied margin = (P − EV_liquid) ÷ P

• Cash-out friction = 1 − (EV_liquid ÷ EV_market)

• P(down) = Σ pᵢ over outcomes where realisableᵢ < P ÷ d

• E[shortfall] = Σ pᵢ × ((P ÷ d) − realisableᵢ) ÷ P(down), over those same outcomes

Assumptions

• The published odds are the odds. The model checks that they sum to 1 and refuses otherwise, but it cannot check that a published table describes the draw that actually runs.
• Draws within one purchase are independent and identically distributed, which is what makes a per-draw figure meaningful. Where a seller guarantees one item of a given tier per purchase, that is not true and the per-draw figures do not apply.
• An independently observed value is a price someone else recorded for the same item, not an appraisal. Its venue, its basis and the day it was observed are published beside it.
• The sell-back rate and the cost of delivery are the ones the seller states. Where either is unstated it is left out and the figure it would have affected is not reported.

Limitations

• No expected value is produced from an incomplete odds table. Missing probability mass does not make a figure slightly uncertain; if the missing mass holds the largest outcome it decides the answer, so no figure is published at all.
• No expected value is produced from declared values alone where independent prices are missing. A catalogue price set by the seller is not a market, and averaging it would restate the seller’s claim as a finding.
• The downside figures describe one draw against the per-draw price. The chance that a whole multi-draw purchase loses money is a different question and is not answered here.
• An observed resale price is a record of past transactions at one venue. It is not a price anyone is obliged to pay, and a thin market can move a long way between observations.
• Nothing here models the time, effort, or minimum thresholds involved in selling an item, beyond the stated sell-back rate and cost of delivery.
• This is an average over the whole distribution. A single purchase is one draw from it, and for a box with a heavy top prize most purchases finish well below the average.

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