A bag holds four stones, each black or white, with every count of black stones from 0 to 4 equally likely. Two stones drawn without replacement are both black. Make a market on a contract paying Rs 100 if the third stone is black, then respond when the interviewer lifts your offer at 80.

Fair value is Rs 75, so 70 / 80 is a fair first market, but a lift at 80 should push your next market up, toward 80 / 90, rather than repeating it. Two black draws give posterior weights 2, 6 and 12 out of 20 on two, three and four black stones, and the third stone is black with probability 0.75. A buyer at 80 may know more; if more than about 17% of such buyers do, selling at 80 loses on average.

A doctor who sees two symptoms does not treat every illness as equally likely any more; she weighs each by how likely it was to produce those symptoms. Here each possible bag is weighted by its prior, one fifth for every count, times the chance it would give two blacks in two draws. With two black stones out of four that chance is 1/6, with three it is 3/6, with four it is certain, and with fewer than two it is zero. The posteriorThe probabilities after updating on the evidence: prior times likelihood, rescaled to sum to one. weights are therefore 2, 6 and 12 out of 20: a 60% chance the bag is all black.

Two stones remain in the bag. If it held two blacks, both remaining are white; if three, one of the two is black; if four, both are. The chance the next stone is black is 2/20 x 0 + 6/20 x 1/2 + 12/20 x 1 = 0.75, so the contract is worth Rs 75. A 70 / 80 market is centred on fair value with Rs 5 of edge on either side, sensible for a contract that pays 0 or 100 and so has a standard deviation of about Rs 43.

Either the buyer is guessing, in which case you sold something worth 75 for 80, or the buyer knows the bag, in which case they lift only when it is all black and the contract is worth 100. Suppose a share q of the people who lift you know the contents; the contract's value given a lift is a weighted mix of 75 and 100, and it crosses your sale price of 80 at q of about 17%. Informed buyers lift only in the all-black world, which is 60% likely; uninformed ones lift half the time whatever the bag holds. At q = 25% the contract is worth 82.1 given the lift; at q = 50% it is worth 88.6. You do not know q, which is exactly why a lift is information.

Move the market up and keep it wide: something like 80 bid, 90 offered. You still believe 75 on the public information, but your offer has just been taken by someone who may know more, and a market maker who keeps selling at 80 to the same eager buyer is the one being arbitraged. Widen before you chase: a lift is evidence that you are wrong or that the other side knows something, and either way the next quote should cost them more. Say it aloud in the interview, because the response to being hit is the part being tested. If the interviewer then sells to you at 80, the lift was noise, and you can drift back toward 75.

The limitation is that the informed-share model is a toy. Real counterparties are partly informed, and their size matters as much as their direction. The reasoning holds: price on the public information, then treat every fill as new information about whoever traded with you.