Optimal Betting Strategy to Reach a Target

Game Theory · Hard · Free problem

A gambler starts with $\

00$ and wants to reach $\$500$. Each bet has probability $p$ of doubling the amount wagered and probability
- p$ of losing it. The gambler keeps playing until reaching $\$500$ or going broke.

Compare three strategies: 1. All-in: Bet the entire current bankroll each round. 2. Half-Kelly: Bet half the current bankroll each round. 3. Fixed bet: Bet a fixed $\

$ each round.

Which strategy maximizes the probability of reaching $\$500$: - (a) when $p = 0.6$ (favorable game)? - (b) when $p = 0.4$ (unfavorable game)?

Hints

  1. Think about how the number of bets affects your chances. More bets help when the odds favor you (law of large numbers), but hurt when odds are against you.
  2. For the fixed-bet strategy, this reduces to the classical gambler's ruin problem with biased coin. Use the standard formula with $q/p$ ratio.
  3. For the all-in strategy, compute how many consecutive wins are needed to go from $\
    00$ to $\$500$. Each round is independent with probability $p$.

Worked Solution

How to Think About It: This is a gambler's ruin problem with a twist -- the choice of bet size fundamentally changes the dynamics. The key principle: when the odds are against you ($p < 0.5$), you want to minimize the number of bets (bet big, let variance help you), because each additional bet works against you in expectation. When the odds are in your favor ($p > 0.5$), you want to maximize the number of bets (bet small) to let the law of large numbers work for you. This is the core of the bold play vs. timid play theorem.

Quick Estimate:

*Case $p = 0.4$ (unfavorable):* - All-in: Starting at $\

00$, you need to double to $\
00$, then $\$400$ -- but $\$400$ to $\$500$ is not a clean doubling. Consider doubling: you need roughly $\lceil \log_2(500/100) \rceil \approx 3$ wins. Probability $\approx 0.4^3 = 0.064$. Not great, but not zero. - Fixed $\ $ bets: You need to gain $\$400$ net, at $-\$0.40$ expected per bet ( \times 0.4 - 2 \times 0.6 = -0.4$). This is a biased random walk that almost certainly hits zero before $\$500$. By gambler's ruin for biased walks, the probability is extremely small.

So all-in wins when $p < 0.5$.

*Case $p = 0.6$ (favorable):* - Fixed $\ $ bets: Expected gain per bet is \times 0.6 - 2 \times 0.6 \cdot 0 = +\$0.40$ (net: win $\ $ with prob 0.6, lose $\ $ with prob 0.4). The drift is positive. By gambler's ruin, the probability of reaching $\$500$ before $\$0$ is high. - All-in: Probability $\approx 0.6^3 \approx 0.216$. Much lower than timid play.

So fixed (small) bets win when $p > 0.5$.

Approach: Use gambler's ruin theory for the fixed-bet case and direct calculation for all-in.

Formal Solution:

Strategy 1: All-in.

Starting at $\

00$, after one win you have $\
00$, after two wins $\$400$. At $\$400$, you cannot bet all-in to reach exactly $\$500$, so assume you bet $\
00$ to reach $\$500$ (or you bet all-in and overshoot). For simplicity, assume the gambler bets the minimum needed when close to the target.

Clean version: starting at $W_0 = 100$, going all-in means $W_{n+1} = 2W_n$ with probability $p$ and $W_{n+1} = 0$ with probability

-p$. To reach $\geq 500$ from
00$, you need at least 3 consecutive wins (
00 \to 200 \to 400 \to 800$). The overshoot does not matter since you stop once you cross $500$.

$P_{\text{all-in}} = p^3$

  • $p = 0.6$: $P = 0.216$
  • $p = 0.4$: $P = 0.064$

Strategy 2: Bet half.

Betting half means you can never go broke (each loss halves your bankroll but never reaches zero). However, you also can never reach $\$500$ in finite bets with certainty -- the path is $W_{n+1} = 1.5 W_n$ (win) or $W_{n+1} = 0.5 W_n$ (lose). After $k$ wins and $j$ losses, $W = 100 \times 1.5^k \times 0.5^j$. You need

.5^k \times 0.5^j \geq 5$.

With $p = 0.6$: the expected log-growth per bet is $0.6 \ln(1.5) + 0.4 \ln(0.5) = 0.6(0.405) - 0.4(0.693) = 0.243 - 0.277 = -0.034 < 0$. The expected log-growth is negative, so the bankroll drifts toward zero in the long run despite the positive expected value. The probability of reaching $\$500$ is less than 1 and can be computed via optional stopping, but it is moderate.

With $p = 0.4$: the log-growth is even more negative, so reaching $\$500$ is very unlikely.

Strategy 3: Fixed $\

$ bet (classical gambler's ruin).

This is a standard biased random walk on the integers. Let the unit be $\

$, so the gambler starts at state $50$ (=
00/2$) and wants to reach state
50$ (= $500/2$). At each step, move up 1 with probability $p$ and down 1 with probability $q = 1 - p$.

The gambler's ruin formula gives the probability of reaching $N$ before $0$ starting from $k$:

$P_k = \frac{1 - (q/p)^k}{1 - (q/p)^N}$

when $p \neq q$, where $k = 50$, $N = 250$.

  • $p = 0.6$, $q = 0.4$: $q/p = 2/3$. Then $(2/3)^{50}$ is astronomically small ($\approx 1.2 \times 10^{-9}$), and $(2/3)^{250}$ is essentially zero. So $P_{50} \approx 1$. With favorable odds, small bets almost surely get you to the target.
  • $p = 0.4$, $q = 0.6$: $q/p = 3/2$. Then $(3/2)^{50} \approx 6.4 \times 10^7$ and $(3/2)^{250}$ is enormous. So $P_{50} \approx (3/2)^{-250+50}/(3/2)^{-250} \approx (2/3)^{200} \approx 0$. Essentially zero -- you will go broke long before reaching $\$500$.

Summary:

| Strategy | $p = 0.6$ | $p = 0.4$ | |----------|-----------|----------| | All-in | 21.6% | 6.4% | | Bet half | Moderate (but < all-in for unfavorable) | Very low | | Fixed $\

$ | $\approx 100\%$ | $\approx 0\%$ |

Answer: - $p = 0.6$ (favorable): Fixed $\

$ bets are optimal -- the positive edge compounds over many small bets, giving near-certain success. All-in wastes the edge by risking ruin. - $p = 0.4$ (unfavorable): All-in is optimal -- you want as few bets as possible to avoid the negative drift. Bold play gives $6.4\%$ vs. essentially $0\%$ for timid play. This is the Dubins-Savage theorem: in a subfair game, bold play maximizes the probability of reaching the target.

Intuition

This problem illustrates one of the deepest results in gambling theory: the Dubins-Savage theorem. When the game is unfair ($p < 1/2$), bold play (betting as much as possible) maximizes the probability of reaching the target. The reason is that every additional bet erodes your expected wealth, so you want to minimize the number of bets and let the variance work for you. When the game is fair or favorable ($p \geq 1/2$), timid play (small bets) is optimal because the positive drift accumulates over many rounds, and the law of large numbers ensures you almost surely reach the target.

This has direct applications in portfolio theory and risk management. A trader with negative expected returns on a position should either take a big concentrated bet or exit entirely -- slowly bleeding through small losses is the worst approach. Conversely, a strategy with a genuine edge should be sized conservatively (Kelly criterion or smaller) to avoid ruin while letting the edge compound.

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