Stopping Problem Variants

Hard3–4 hrsCasino StoppingBayesian UpdatingPrice TargetWalk Away

Overview

Real-world stopping problems go beyond textbook formulations. In a casino with a fixed bankroll and target, the optimal strategy is to BET BIG (maximize variance) since you're playing against a house edge — more bets means more time for the edge to grind you down. This is the opposite of the optimal strategy for a FAVORABLE game. The key principle: in an unfavorable game, bet big and hope to get lucky. In a favorable game, bet small (Kelly fraction) and grind out the edge. Understanding these variants shows you grasp both the mathematics and the economics.

How to Recognize

→Casino-style: 'how long should you play before walking away?'
→Price-target problems: 'your stock hits X, should you sell?'
→'Play until you've won K times, then stop' — negative binomial
→Stopping rules based on belief updates (Bayesian stopping)

Step-by-Step Approach

1.Casino stopping: set a WIN TARGET and LOSS LIMIT; compute optimal ratio
2.Price target: threshold = E[future returns] - opportunity cost
3.Bayesian: update P(state) after each observation, stop when posterior is decisive
4.Kelly criterion: size bets optimally, walk away when edge disappears

Key Formulas

P(\text{hit win target before loss limit}) = \frac{1-(q/p)^k}{1-(q/p)^N}

(Gambler's Ruin)

\text{Kelly fraction: } f^* = p - q/b

where

b

= net odds,

p

= win prob

E[\text{time to ruin} | \text{fair game, start}=k] = k(N-k)

\text{Optimal casino strategy: maximize } P(\text{win target}) \times \text{win} - P(\text{loss}) \times \text{loss}

Worked Examples

0Quick Example

Casino: bankroll

100, target

200, bet $10 on roulette (p=18/38). P(success) ≈ 11.7%. If edge is -2.7% per bet, play faster (more bets = more house edge extracted). Best strategy: maximize variance.

Problem

You enter a casino with $100 and want to reach$ 200. The game is roulette (red/black, p=18/38). You can bet any amount each round. What bet size maximizes P(reaching $200)?

Solution

This is a biased gambler's ruin with p < ½. The house has an edge.

P(reaching N | start k, bets = b per round): higher b means fewer steps → less exposure to negative EV per step.

EXTREME STRATEGY: Bet everything at once. P(win) = p = 18/38 ≈ 47.4%. One round.

GRADUAL: Bet $10 each time. P(reaching$ 200) = (1-(q/p)^{10})/(1-(q/p)^{20}) where q/p=20/18.

(20/18)^{10} ≈ 2.867, (20/18)^{20} ≈ 8.22. P = (1-2.867)/(1-8.22) = (-1.867)/(-7.22) ≈ 25.9%.

Bet everything: P = 18/38 ≈ 47.4% >> 25.9%.

CONCLUSION: In an unfavorable game, MAXIMIZE bet size to minimize expected rounds of play.

Answer

Bet everything ( $100) on one spin: P(success)=47.4%. Gradual betting ($ 10/round): P≈26%. For UNFAVORABLE games, maximize bet size to maximize P(hitting target).

Common Mistakes

!
For unfavorable casino games: betting small to 'last longer' actually HURTS you (gives house more edge extraction). Bet big.
!
Confusing 'maximize P(win)' with 'maximize EV'. In a negative-EV game, P(win) can be increased by maximizing variance, but EV is always negative.
!
Forgetting that the Kelly criterion assumes POSITIVE expected value. Never use Kelly in negative-EV games.
!
In price-target stopping: treating the problem as static when the distribution of prices changes over time (non-stationarity).

Practice Problems

Click "Show Answer" to reveal

You play a FAIR coin flip game ( $1 per flip). Start with$ 10. You stop at $15 (win) or$ 0 (bust). What is P(winning) and E[# flips]?

Hint: Gambler's ruin fair: P(win) = k/N = 10/15 = 2/3. E[duration] = k(N-k) = 10×5 = 50.

In a fair game, you start at $k = 50 and must stop at$ 0 (ruin) or $100 (win). You can choose your bet size:$ 1, $5, or$ 25 per round. Does bet size matter?

Hint: For FAIR games, P(win) = k/N regardless of bet size (as long as it divides evenly). E[duration] scales with 1/bet_size².

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