The interviewer asks for a market on the maximum of four fair dice, then lifts your offer twice in a row at 5.4. What is the contract worth, how wide should the market be given its skewed distribution, and how do you requote?

Fair value is 5.24, but half the probability sits on 6, so the risk is lopsided: an offer can lose at most the gap to 6, a bid can lose almost 4. Open at 4.9 bid, 5.4 offer, the bid further from fair than the offer. Two lifts from someone who has seen the dice say the max is probably 6: if half your counterparties are informed, the chance of 6 rises to 91% and fair value to 5.85. Requote 5.7 at 6.0.

The trick is to count the complement. The maximum is at most k only if all four dice are at most k, which has probability (k/6) to the fourth. So P(max = k) is (k/6)^4 minus ((k-1)/6)^4, which puts 51.8% on 6, 28.5% on 5 and only 6.2% on 3 or below. Weighting gives an expected value of 5.245, with a standard deviation of 0.95. Note that the median and the most likely value are both 6, while the mean is pulled down to 5.24 by the long tail of low outcomes.

Think of the two sides separately. Selling at 5.4 can lose at most 0.6, because the contract cannot settle above 6; buying at 4.9 can lose up to 3.9 if all four dice come up 1. Against a counterparty who has seen the dice, the side with the bigger worst case needs the bigger cushion, so the bid sits 0.34 below fair and the offer only 0.16 above. A symmetric market of 5.0 at 5.5 would look tidy and give a seller who saw three low dice a far better trade than a buyer who saw a six. The width of 0.5 is about half a standard deviation, tight enough that an uninformed player will trade.

Someone who has seen the dice buys at 5.4 only if the max is 6, since 5 or less settles below 5.4. Model the interviewer as informed half the time and trading at random otherwise. A lift then happens 75% of the time when the max is 6 and 25% of the time when it is not, so each lift triples the odds on a six. The prior odds are 0.518 to 0.482. After one lift the chance of a six is 76% and fair value is 5.63; after two it is 91% and 5.85, using 4.43 as the average maximum when it is below 6.

So requote 5.5 at 5.8 after the first lift and 5.7 at 6.0 after the second. An offer at 6.0 cannot lose, because the contract can never settle above it, which makes it the natural place to stand when the evidence points to a six. You are short two at 5.4 and marked at about 5.85, down about 0.9: the price of having learned. The limitation is the model: the 50% informed share is a guess, and an interviewer who lifts to test your nerve would make the update smaller. Say the assumption out loud; the interviewer is grading how you update, not the decimal.