Market Making Card Bet Game In an interview I received the follow question: We have 3 cards face down, and we give each card in a deck of 52 a numeric score ( A = 1, 2=2, .... , J=11, Q=12, K = 13). The interviewer asked me to find the expected sum of these 3 cards, I approximated that each card has an expected value of (1+2+...+13)/13 = 7, and so the sum is approximately going to be 21 (or adjusted down slightly to 20.5). Then the game begins as follows.
You begin with $1000. I will quote a price to play this game, and you are able to either buy x units, or sell me x units, then we turn over the cards and you realize your profit or loss. For example:
Lets say you are given a price of 25, clearly this is greater than 21, so I would choose to sell (Short) 10 units at that price. The cards turn out to have a sum of 23, so I make a profit of (25-23)*10 = 20. So my total money now is 1020, and so on.
We played several rounds of this, and I was just wondering what is an appropriate strategy for this game? I tended to increase the units I would buy when quoted a very low price, and increase the number I would sell when quoted a very high price (relative to 21), but now that I think about it, is there a more sophisticated way of approaching this problem?
 A: Mathematically, the problem is a bit ill-defined.


*

*We're not certain of your goal: Do you care about your strategy's expected value of winnings only? Or do you want to end up above $2000 at the end of the interview but are very averse to busting? Etcetera.

*We're also not certain what sorts of prices you're going to be given in the future.
That said, it makes sense to bet more when the price is very high or very low, as you're more certain to win in those cases, so your risk is lower, and furthermore your expected return is higher.
How to go about this in a coherent manner? I think there are several ways. One I'll mention is to have some sense of "acceptable risk." For example, you could decide that on any given round, you don't want more than an $X\%$ chance of losing more than $Y\%$ of your holdings, i.e. use a value at risk metric for deciding how much to wager on any given round.
From a pure expectation point of view, you just buy/sell as much as you can each round.
A: Another way to look at it. It is a uniform distribution which has a mean of $$ (max+min)/2 $$
The range of outcomes are (3,39) so the mean is:$$ (39+3)/2 = 21 $$
You could even calculate the standard deviation which is  given by:
$$ (max-min)/2\sqrt{3} $$
$$ (39-3)/2\sqrt{3}\approx10.3923 $$
Your strategy would be to increase units when price is outside one standard deviation so anything less than 10.6 or greater than 31.4.
