Betting Strategy on 45/55 game with free money that restricts cash out

Question

Suppose you have 1000 and are playing a betting game where you have 45% chance of winning your betted amount and 55% of losing your betted amount (e.g. bet 500, win or lose that amount). If your opponent was to offer you an additional 1000 to your betting pool on the requirement that you can not cash out with the extra 1000 until your total balance exceeds 10000, would you take the offer? If you don't take the offer, would you play with your original 1000 (and walk away at any time) or walk away without playing at all?

Answer 1

This might not be an "optimal strategy." But, it does prove that this is a positive expected value proposition.

When the game is against you, to maximize expected value, minimize the total number of bets placed.

So, attempt to double-or-nothing on every bet. To reach 10,000 you will need to double-up 3 times.

You have a $0.45^3 = 0.091$ chance of walking away with $\$16,000$ which is more than you currently have.

A secondary question is... what is your risk aversion? If you are risk averse, (and most people are) you will need something more than positive expected value to take a long-shot bet with a small positive expected value.

Answer 2

You did not say what the minimum bet was. For example: If minimum bet is 1, then with a total pool of 2000, you can afford 2000 trials in the absolute worst case. If the minimum bet is 0.1, then you can afford 20000 trials in the worst case. If the minimum bet is 0.01, then you can afford 200,000 trials in the worst case.

If there is no minimum bet, this becomes an unlimited trials experiment since you can adopt the strategy of halving your bet amount without making it zero until you recover losses. You can start with any bet amount that is less than 1000. This strategy allows you to stay in the game infinitely and is guaranteed you won't lose your money that you start with (1000).

Note that both players can adopt this strategy of staying in the game.

If your opponent is offering you 1000 with a cashout restriction, but doesn't prescribe a minimum bet, it is a positive expected value proposition due to the unlimited trials. So, you should take it.

Your opponent can cash out anytime whereas you can't until you reach the cashout threshold. If that happens, you still have a total funds > 1000 because you never bet your own money!

Alternate scenario (with minimum bet): If there is a minimum bet, you should probably walk away. The reason is you have only a finite number of trials in order for you to meet the cashout threshold. Since you cannot abandon the game without exhausting your bank roll, you have risk of losing it all.

Answer 3

It can be shown that an optimal strategy in this game is "bold play." That is, when your bankroll is less than or equal to $5000$, bet it all. When it is more than $10,000$, bet just enough to reach $10,000$ if you win, retaining the rest as a reserve. See "How to Gamble if You Must" for an analysis. (Note that there are infinitely many optimal strategies, but bold play is the simplest to describe.)

Let $p(k)$ be the probability of reaching $10000$ if your current bankroll is $k$. Then we want to know whether $10000p(2000)>1000$, that is, whether $p(1000)>.1$

Using bold play, we have $$\begin{align}p(2000)&=.45p(4000)\\ &=.45^2p(8000)\\ &=.45^2(.55p(6000)+.45)\\ &=.45^3+.45^2\cdot.55(.45+.55p(2000))\\ &=.45^3\cdot1.55+.45^2\cdot.55^2p(2000) \end{align}$$ so that $$ p(2000)=\frac{1.55\cdot.45^3}{1-.55^2\cdot.45^2}\approx0.15046$$

and you should accept the proposition.