# Optimizing a Percentage of Wealth for Gambling Random Walk

Okay, so I am curious about a problem.

Say that you are given the opportunity to make a certain bet in which $P(Win)=.52$ and $P(Lose)=.48$ and that their payouts are equal but opposite. You will repeat the bet $10000$ times. Assuming that you start with $1000$ units, and you are allowed to bet $p$ percent of your total wealth after each round. For example, if you choose $p=.1$ you would bet $100$ units the first round. If you win the first round, you would bet $110$ units the next time. Conversely, if you lost the first round, you would bet $90$ units the second round. Notice that you can never go completely bankrupt since you are only betting a predetermined percent of your wealth each time.

Assuming that you can only, control $p$, what percent of your wealth would you choose to bet? I.e. what value of $p$ would give you the highest expected wealth after $10000$ trials?

My attempt: When I saw this problem, I initially thought that a larger $p$ would surely increase the expected value at the end of $10000$ rounds since you had a greater than $.5$ probability of winning.

After struggling to come up with a formula, I built a simulation and found out that this was clearly not the case. (Any simulation with $p\gt.1$ resulted in a near-zero end value).

This got me thinking and I thought about this problem as $(1+p)^{5200}*(1-p)^{4800}$ since there would be expected $5200$ wins and expected $4200$ losses. I then differentiated and looked for a $0$ over the interval $(0,1)$ to find a maximum. I ended up getting a derivate of $5200(1-x)^{4800}(1+x)^{5199} -4800(1-x)^{4799}(1+x)^{5200}$ and end up with a value of $.04$

When I plug in $.04$, it does indeed yield a higher end value, but it is so variable from time to time it is difficult to tell if my answer is correct.

I am hoping that someone can provide guidance on whether my general approach to this problem is correct, or if there is some other way to solve such a problem?

• What is the objective? Is it to maximize expected final wealth? If so, one should bet $100$% of current wealth each time (i.e., $p=1$). – quasi Jul 13 '17 at 3:26
• Yes, objective is to maximize expected final wealth. If one were to bet $100$% of wealth, just one loss would result in running out of money and at least one loss is all but guaranteed in $10000$ trials – user345 Jul 13 '17 at 3:32
• That has nothing to do with expected wealth. The expected winnings are .02 units for every unit bet. So the more units bet, the higher the expected winnings. With the $100$% strategy, bankruptcy is almost certain, but the expected wealth is maximized. – quasi Jul 13 '17 at 3:33
• @quasi maybe I am missing something, but if you were to bet $100$% of wealth, one loss would use all of you wealth and you would be unable to continue on. Thus, you would be almost certain to end up with $100$% loss. I agree that $100$% would give highest expected value after $1$ round, but I don't see that it would be possible to maximizing for $10000$ rounds – user345 Jul 13 '17 at 3:36
• Expected values add. – quasi Jul 13 '17 at 3:37

Assuming the objective is to maximize expected final wealth, one should bet $100$% of current wealth each time (i.e., $p=1$).
The expected winnings are $.52(1) + .48(-1) = .04$ units for every unit bet, so the more units bet, the higher the expected winnings.
Note: With the $100$% strategy, bankruptcy is almost certain, but the expected wealth is maximized.
Using the $100$% strategy, the expected wealth after $n$ bets is $$(.52)^n\left(2^n\right)(1000)$$
For small $n$, say $n \le 10$, you can easily verify this with a simulation.