You and three computer quoters make markets on the sum of five cards drawn from one suit numbered 1 to 13. Each bot has seen one card and its mid is 37, 33 or 39. What did each bot see, and where do you quote?

The bots have seen a 10, a 4 and a 13, and the contract is worth about 39.8. Each bot's mid is (2c + 91)/3, so the card is (3 x mid minus 91)/2. The three known cards sum to 27; the last two come from the ten cards left, averaging 6.4. Quote around 38.8 at 40.8 and buy the offers at 34 and 38.

Start with one bot. It knows its card c exactly and knows the other four come from the twelve cards it has not seen, which sum to 91 minus c. Its fair value is c plus four times the average of what is left: c + 4(91 - c)/12, which is (2c + 91)/3. A card of 7 gives 35, the same as knowing nothing, which is a useful check. Each point on the card moves the mid by only two thirds of a point, because a high card seen is a high card that cannot be among the unseen four. It is the same logic as guessing a class's total marks after hearing one student's score: a top score raises the total, but it also means that top score is no longer available to anyone else.

Invert the rule: c = (3m - 91)/2. Mid 37 gives 10, mid 33 gives 4 and mid 39 gives 13. They are three different cards, which is a consistency check worth saying aloud; if the inversion gave two 10s you would know one bot was shading its quote. With 10, 4 and 13 known, only two cards remain unknown, drawn from the ten left, which sum to 64 and average 6.4. The contract is worth 27 plus 2 x 6.4, which is 39.8. You have pooled three private signals that no single bot has, and that is the whole edge in this game.

The two unknown cards add uncertainty with a standard deviation of about 4.7 points, so the contract is far from settled. Centre your quote on 39.8 and keep it about as tight as the bots, 38.8 at 40.8, because you now know more than any of them. Then look at their prices. The card-4 bot offers at 34, nearly 6 below value, and the card-10 bot offers at 38. Both are selling on one card of information when the pooled answer is 39.8. Lift both offers. The card-13 bot's offer of 40 is 0.2 above value, so leave it.

Say the limit before the interviewer does. The inversion multiplies any error in a mid by 1.5: if a bot shades its quote by one point to manage its own inventory, your inferred card is off by 1.5 and your fair value by about 1.3 once the averaging is redone. So treat each inferred card as an estimate, and watch whether a bot's mid moves after it trades; a quote that drifts with its position is telling you about inventory, not cards. After you buy, you are long, so shade your own quote down a little to attract sellers and lighten up.