The informed doorman

A fill is bad news about value. When a seller who knows the true value accepts your bid, you learn that the value was below your bid. Price against the conditional expectation given a fill, E[V | fill], never against the prior mean. For a value uniform over integers 1 to 100 with a bid of 60, the seller accepts exactly when V is 60 or below, so the filled value averages (1 + 60) / 2 = 30.5 while you pay 60, for an expected profit of 30.5 - 60 = -29.50.

Words you need

Bid and ask: The bid is the highest price a buyer is offering right now; the ask (also called the offer) is the lowest price a seller will take. To sell immediately you sell at the bid; to buy immediately you buy at the ask.e.g. A market '99 at 101': bid 99, ask 101.
Fill: Your order actually traded. 'My bid got filled' means someone sold to me at my bid.
Expected value (EV): The average result if you could repeat the bet many times: each outcome times its probability, added up.e.g. A die pays its face in dollars: EV = 3.5.

Why it works

The seller accepts exactly when the value is below your bid, so the filled distribution is the prior truncated above at the bid.

E[V | fill] = (1 + B) / 2, Profit = E[V | fill] - B = (1 - B) / 2

Spot it: A counterparty who knows the value only trades when your price favors them.

Technique reference and sources

Picture a doorman who only lets you buy when you overpay. Every bid below the true value is turned away, and every accepted bid was too high. Your average purchase is therefore the average of all values your bid could beat, which is always far below your bid.

When this method fits

A counterparty who knows the value only trades when your price favors them.
You bid on an asset with unknown common value, such as a sealed-bid auction.
An interviewer asks for expected profit conditional on winning or being filled.
You need the break-even bid given a private resale benefit on top of common value.

Write the acceptance rule. The seller accepts exactly when V <= B. Fills only come from values at or below your bid.
Condition the distribution. Given a fill, V is uniform over integers 1 to B, averaging (1 + B) / 2.
Subtract the price paid. Expected profit is E[V | fill] - B = (1 - B) / 2, which is negative for every bid above 1.
Add any private benefit. With a resale benefit c per unit, profit is (1 + B) / 2 + c - B. When 1 + 2c does not exceed the top value M, the largest safe integer bid is floor(1 + 2c); otherwise the bid fills every value and the largest safe integer bid is floor((M + 1) / 2 + c); a bid above the top value M can still be safe, but bidding M earns more.

Uniform 1 to M fill conditioning: With V uniform over integers 1 to M and a bid B within range, E[V | fill] = (1 + B) / 2 and expected profit = (1 - B) / 2.

M = 100, B = 60: value 30.5, profit -29.50
M = 50, B = 20: value 10.5, profit -9.50

Private-benefit break-even: With a private benefit c per unit on top of common value, expected profit is (1 + B) / 2 + c - B. When 1 + 2c is at most the top value M, the largest safe integer bid is floor(1 + 2c); otherwise it is floor((M + 1) / 2 + c); a bid above M can still be safe, but bidding M earns more.

c = 15 on 1 to 100: max safe bid 31, since 16 + 15 - 31 = 0
c = 10 on 1 to 100: max safe bid 21, since 11 + 10 - 21 = 0
M = 20, c = 10: 1 + 2c = 21 exceeds M, so max safe bid is floor(10.5 + 10) = 20

Common traps

Pricing the bid against the prior mean, computing 50.5 - 60 = -9.50 on a 1 to 100 value. The seller only accepts when V is 60 or below, so the relevant mean is 30.5, not 50.5. The true expected profit is 30.5 - 60 = -29.50.

Treating the fill probability as irrelevant and reporting only the conditional profit as the ex ante profit. Conditional profit applies once filled. Ex ante, multiply by the fill probability B / M: with M = 100 and B = 60, ex ante profit is 0.6 × (-29.50) = -17.70.

Bidding the prior mean plus the private benefit without conditioning. Add the benefit to the conditional value, not the prior mean. With benefit 15, bidding 60 gives 30.5 + 15 - 60 = -14.50, still a loss.

Another example: Bid 40 with a private resale benefit of 15

A fill means V is uniform over integers 1 to 40
Conditional common value: (1 + 40) / 2 = 41 / 2 = 20.5
Your private value per unit: 20.5 + 15 = 35.5
Expected profit: 35.5 - 40 = -4.50

Source notes

These explanations and questions are written for QuantArc. Links show the underlying mathematics and learning design.

Jane Street: Probability & Markets Trading Interview Guide
Interview preparation guide (page 9) testing optimal stopping and market making on a d6 with one optional reroll. Lesson examples and questions are original.

IES: Organizing Instruction and Study to Improve Student Learning
Supports worked examples, retrieval, explanations and spaced study. Does not establish QuantArc's exact lesson length, mastery threshold or review intervals.