Gambler’s Ruin

Gambler’s Ruin

Consider a gambler who starts with an initial fortune of $1. On each successive gamble, the gambler either wins $1 or loses $1 independently of previous outcomes, with probabilities \(p\) and \(q = 1 - p\), respectively.

Suppose the gambler begins with a kitty of $A and places bets against a “Banker” who has an initial fortune of $B. The game is viewed from the gambler’s perspective. By convention, the Banker is the wealthier of the two.

Key parameters:

• Probability of a successful gamble: \(p\)
  
• Probability of an unsuccessful gamble: \(q = 1 - p\)
  
• Success-to-failure ratio: \(s = \frac{p}{q}\)
  
• Typically, the game is biased in favor of the Banker (i.e., \(q > p\), so \(s < 1\))

Let \(R_n\) denote the gambler’s total fortune after the \(n^\text{th}\) gamble.

• If the gambler wins the first game: \(R_1 = A + 1\)
  
• If the gambler loses the first game: \(R_1 = A - 1\)

The total amount at stake is the Jackpot, \(A + B\). The game ends when the gambler either: - Wins the entire jackpot: \(R_n = A + B\)
- Loses everything: \(R_n = 0\)

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Simulating a Single Gamble

To simulate one bet, generate a random number between 0 and 1:

runif(1)

## [1] 0.5262137

Assuming the game is biased in favor of the Banker:

p <- 0.45
q <- 0.55

If the random number is less than 0.45, the gambler wins; otherwise, the Banker wins.

Example outcomes:

runif(1)  # [1] 0.1251274  → Gambler wins
runif(1)  # [1] 0.754075   → Banker wins
runif(1)  # [1] 0.2132148  → Gambler wins
runif(1)  # [1] 0.8306269  → Banker wins

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Tracking the Gambler’s Fortune Over Time

Let \(A\) be the initial kitty, and \(B\) be the Banker’s wealth. We’ll keep track of the gambler’s fortune using a vector \(R_n\):

A <- 20
B <- 100
p <- 0.47
Rn <- c(A)

probval <- runif(1)

if (probval < p) {
  A <- A + 1
  B <- B - 1
} else {
  A <- A - 1
  B <- B + 1
}

Rn <- c(Rn, A)

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Simulating Until the Game Ends

The loop below runs until the gambler either loses all his money or wins the entire jackpot. A break statement ends the game if the gambler wins:

A <- 20
B <- 100
p <- 0.47
Rn <- c(A)
Total <- A + B

while (A > 0) {
  UnifVal <- runif(1)

  if (UnifVal <= p) {
    A <- A + 1
    B <- B - 1
  } else {
    A <- A - 1
    B <- B + 1
  }

  Rn <- c(Rn, A)
  
  if (A == Total) break
}

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Simulating Multiple Games to Examine Duration

We can simulate many games and record how long each lasts (i.e. how many rounds until the gambler goes broke or wins the jackpot):

A.ini <- 20
B <- 100
p <- 0.47
M <- 1000
RnDist <- numeric()
Total <- A.ini + B

for (i in 1:M) {
  A <- A.ini
  Rn <- numeric()
  Rn[1] <- A
  
  while (A > 0) {
    UnifVal <- runif(1)
    
    if (UnifVal <= p) {
      A <- A + 1
      B <- B - 1
    } else {
      A <- A - 1
      B <- B + 1
    }
    
    Rn <- c(Rn, A)
    
    if (A == Total) break
  }
  
  RnDist <- c(RnDist, length(Rn))
}

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Distribution of Game Durations

Now you can analyze or plot RnDist to explore the distribution of game durations—i.e., how many rounds typical games last before ending.