We are going to play a game. There are 3 unknown cards (for each one, values are distributed uniformly between 1 and 13). I am going to make a market on the sum of the 3 cards, and then you must make a trade. The market that I present to you is I am willing to buy at 17 and sell at 19. How much would you be willing to pay for me to reveal the first card before we trade?
Here is my thought process.
The expected value for each card is (1+13)/2 = 7. Hence the expected sum for 3 unknown cards is 21. As the initial expectation is 21, the initial expected profit is 2, as you can buy at 19.
If you choose to reveal a card, there is a 1/13 chance that it is any of the 13 values. The expectation for the remaining 2 cards will still be 7 + 7 = 14. For instance, if a 1 is revealed for the first card, the expected sum is now 15 and the profit would be 2, as the market is 17 at 19. As such, the expected profit after we reveal the first card is (averaging profits for cases when first card is 1,2,3...,13):
$\frac{1}{13}(2 + 1 + 0 + (- 1) + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 +8) = \frac{38}{13}$.
Notice that the in the above equation, the profit is 0 in the case when the first card is a 3 or a 5 as then the expected sum is 17 and 19 respectively. Similarly, you will get a -1 profit for when the first card is a 4 and hence the expected sum is 18, so trading on either side will result in a 1 loss.
My expected profit improved by $\frac{38}{13} - 2 = \frac{12}{13}$, so that is how much I am willing to pay to reveal the first card. Is my reasoning correct?