Three players are each dealt one card from 1 to 10 without repeats, and the contract settles at the highest card. You hold a 7. Price it, then reprice after another player immediately bids 8.5.

On your 7 alone the contract is worth about 8.22; after the 8.5 bid, read as a 9 or a 10, it is worth about 9.56. A player quoting a point wide bids 8.5 only if they value it near 9.5, which fits a 9 or a 10 and not an 8. Pull any offer below 9.5 at once, quote around 9.0 at 9.8, and do not sell to the bidder.

The other two cards are a random pair from the nine you cannot see, so there are 36 equally likely pairs. Count where the highest card lands. Both below your 7 means both from 1 to 6: 15 pairs, and the answer is 7. Otherwise the highest card is the bigger of the two: 10 appears in 8 pairs, 9 without 10 in 7, 8 without 9 or 10 in 6. The expected highest card is (7 x 15 + 8 x 6 + 9 x 7 + 10 x 8) / 36 = 296/36, about 8.22. With no other information, you would quote around 7.7 at 8.7.

Think of an auction where one bidder opens well above the estimate before anyone else speaks: the room learns that this bidder knows, or believes, something good about the lot. A bid is a statement about the bidder's card, and you can invert it the same way you would invert a price from a pricing rule: compute what the contract is worth to each possible hand and ask which hands would bid 8.5 with an edge. To a holder of an 8 it is worth 8.64, to a 9 9.22, to a 10 exactly 10. Players in this game quote about a point wide, so their bids sit 0.5 to 1.5 below their value; a bid of 8.5 implies a value between 9.0 and 10.0. Only the 9 and the 10 fit. An 8 would be bidding with an edge of 0.14, which nobody in a one-point-wide game does on the first call.

Condition on it. Given your 7, the bidder is equally likely to hold the 9 or the 10. If the 10, the contract settles at 10. If the 9, it settles at 10 only if the third player has the 10, one chance in 8, and otherwise at 9, so it is worth 9.125. Averaging the two, the contract is worth about 9.56, up from 8.22: one bid moved fair value by 1.34 before a single trade. If you are less sure of the reading and let the bidder hold an 8 as well, the value is 9.17; either way, the outcomes 7 and 8 have all but left the distribution.

Move first. If you had an offer out at 8.7 from your own card, it is now about 0.86 below value; the bidder will lift it, so pull it before saying anything else. Requote around 9.0 at 9.8, a market that brackets both readings of the bid, and do not sell to the bidder at 8.5. The trade worth doing is with the third player: if they have not reacted and still quote around their own card, they may offer near 8.5 to 9, and buying there is buying what the room now knows is worth more. Your own 7 is now nearly irrelevant to the settlement, which is the point interviewers are looking for: your card mattered most before the bid; the bid matters more than your card after it.

Say the limit: the bidder could be bluffing. A bluff is expensive, though. A holder of a 3 who bids 8.5 and gets hit pays 0.83 a lot for the lie, and a group game repeats, so a player who bluffs often stops being believed. Weight the bluff low but not at zero, and keep size small until a second piece of information, a trade or another quote, confirms the reading.