# Bet sizing problem

Discussion in 'Scientific Statistics Math' started by Luna Moon, Dec 2, 2010.

1. ### Luna MoonGuest

Hi all,

Lets say we are facing the following gambling and bet sizing problem.

In a series of N bets, if we keep betting using \$1, the outcomes are
realizations of a random variable following certain distribution.

That's to say, assuming constant bet size \$1, the outcomes x_1,
x_2, ... x_N follow a random variable.

Now we want to optimize the bet sizes. Instead of the constant bet
size \$1, we have a series of bet size b_1, b_2, ..., b_N. And
correspondingly the new outcomes are y_1, y_2, ..., y_N.

However, due to timing and the operational issues, there is a delay in
outcomes.

And the rule is as follows:

Lets say today is the i-th day.

When you place a bet with size b_{i}, you have already observed the
outcome y_{i-1}; however, your bet b_{i} will be effective tomorrow,
and the corresponding outcome will occur as y_{i+1} tomorrow.

Therefore, the correspondence is y_{i-1} => b_{i} => y_{i+1}.

Notice that b_{i} doesn't influence y_{i} and y_{i} is impacted by
b_{i-1}, so on and so forth.

Now the questions are:

1) Using the empirical histogram/distribution of the length of
consecutive wins and losses (win: positive y_{i}; loss: negative
y_{i}), is there a way to devise a look-up table to place the bets
optimally?

2) The above question has neglected the magnitude of wins and losses.
So if we use not only the empirical histogram/distribution of the
length of consecutive wins and losses, but also use the empirical
histogram /distribution of the outcomes, is there a way to optimize
the bet sizes?

3) In addition to the above questions, if we not only want to optimize
the outcomes, but also want to contrain the standard deviation of the
outcomes y's. Is there a good way to do such optimization?

Thanks a lot!

Luna Moon, Dec 2, 2010