Five players trade a contract that settles at the sum of ten dice rolled one at a time in public. After four rolls totalling 17, one player bids 38 and another offers 36. What is fair value, and what can you lock in by trading between them?

Fair value is 38, and buying 5 lots from D at 36 while selling 5 to B at 38 locks 2 points a lot, Rs 1,000, with no position left. Six dice remain, each worth 3.5 on average, so 17 + 21 = 38. The two quotes cross, a bid above an offer, so you take both sides at once. The profit does not depend on the dice or on your fair value being right.

Split the sum into what you know and what you do not. The 17 already rolled is fixed; each of the six dice still to come adds 3.5 on average, so fair value is 17 + 6 x 3.5 = 38. It is the same arithmetic as a cricket chase: the runs already on the board are certain, and you add the average for the overs left. The uncertainty is only in those six dice: their total has a variance of 6 x 35/12 = 17.5, a standard deviation of about 4.2 points, so the final sum will usually land between 34 and 42.

Look at the prices, not at your fair value. Player B will pay 38 and player D will sell at 36. When one player's bid is above another player's offer, the market is crossedA state where the best bid is higher than the best offer. Anyone can buy at the offer and sell at the bid for an immediate profit., and you can buy from one and sell to the other with no view on the dice. Buy 5 lots from D at 36, sell 5 lots to B at 38. Your position is zero, so the last six dice cannot hurt you, and you keep 2 points on each lot: 2 x 5 x Rs 100 = Rs 1,000.

Notice what the fair value was used for. It tells you that D's offer is cheap and B's bid is exactly fair, but the locked trade would be worth 2 points even if your fair value were wrong. That is the difference between an edge and an arbitrage: an edge pays on average; an arbitrage pays every time.

In a game where every roll is public, nobody has private information, so the cross is not a signal about the dice. It usually means a player is managing a position rather than pricing the contract. Player D may be long 20 lots and keen to sell; player B may be short and keen to buy back. Listen for that in a group game, because it tells you who will keep leaning on the market. Speed matters too: the first player to hit both quotes takes the profit, so say both trades in one breath.

Suppose D offers 10 lots at 36 and B bids for only 5 at 38. The first 5 lots are the locked trade as before. The extra 5 lots bought at 36 are an edge, not an arbitrage: worth 2 points each on average, Rs 1,000 in total, but with a standard deviation of about Rs 2,092 from the dice. You would take them if your risk limit allows, then show a market of your own around 38, perhaps 37 bid at 39 offered, to sell them back to the room.