Long-term behavior of casino games

Technical Summary: Long-term behavior of casino games

Problem Statement
The paper addresses the asymptotic behavior of the ratio of total return (or total profit) to the total amount bet in casino games. While the strong law of large numbers guarantees that this ratio converges almost surely to the Return to Player (RTP) or negative House Advantage (HA) when wagers are independent and identically distributed (i.i.d.), this assumption fails to model realistic casino scenarios. In practice, bet sizes often vary over time and may depend on past outcomes (e.g., betting systems), and games may involve compound structures (multiple bets per round) or future dependence (bets resolved over multiple spins or rolls, such as in craps or roulette with the en prison rule). The authors seek a general framework to determine the long-term limits of these ratios under more realistic, non-i.i.d. conditions, relying only on the intrinsic parameters of the game.

Methodology
The authors establish a general probabilistic framework based on a sequence of wagers $(B_n, R_n, X_n)_{n \ge 1}$, representing bet, return, and profit, adapted to a filtration $(\mathcal{F}_n)_{n \ge 0}$. The core of the methodology relies on two main mathematical tools:

1. Martingale Theory: Specifically, Chow's (1967) strong law of large numbers for martingales, which allows the authors to show that the difference between a sum of random variables and the sum of their conditional expectations converges to zero almost surely, provided second moments are bounded.
2. Doob's Decomposition: The authors decompose the cumulative sums of bets, returns, and profits into a martingale component (the deviation from conditional expectations) and a predictable component (the sum of conditional expectations).

The main result (Theorem 3.1) demonstrates that under mild conditions—specifically, bounded second moments and a non-vanishing average bet size—the long-term ratio of total returns to total bets behaves asymptotically like the ratio of the sums of their conditional expectations. This allows the replacement of random variables with their conditional expectations given the past, which are often deterministic or easier to analyze.

Key Contributions and Results
The paper derives limit theorems for four distinct classes of casino games, showing that the long-term behavior is governed by intrinsic game parameters (RTP and HA) rather than the specific betting strategy, provided the strategy does not violate the boundedness conditions.

1. Simple Games (Sections 4 & 5):

   • For games with one or multiple bets per coup where outcomes are resolved immediately, the authors prove that if the underlying unit bets are i.i.d., the long-term profit-to-bet ratio converges almost surely to $-HA$ (or RTP to 1).
   • In games with multiple betting opportunities (e.g., roulette with different bet types), the limit is bounded by the minimum and maximum RTP/HA of the individual opportunities. If the game is "HA-invariant" (all opportunities have the same HA, as in standard single-zero roulette), the limit is unique regardless of the bettor's allocation.

2. Compound Games (Sections 7 & 8):

   • The framework extends to games where additional bets are made during a coup (e.g., doubling/splitting in blackjack, free odds in craps, Three Card Poker).
   • The authors define a "compound wager" where the total bet is a base bet multiplied by a random factor determined within the coup. They show that the long-term ratio converges to the RTP/HA calculated over the total expected bet of the compound wager, not just the base bet.
   • This resolves ambiguities in literature regarding whether to calculate house advantage based on the initial bet or the total amount risked.

3. Games with Future Dependence (Section 9):

   • For games where bets are not immediately resolved (e.g., craps with come bets, roulette with en prison), the authors introduce a "round-based" filtration. A round is defined as a sequence of spins/rolls ending at a stopping time where all pending wagers are resolved.
   • Theorem 9.1 establishes that if the round lengths have bounded second moments, the asymptotic behavior of the round-based totals converges to the same limits as the spin-based totals, provided the conditional expectations of returns per unit bet remain bounded.

4. Application to the Leigh Roulette System (Section 6):

   • As a concrete application, the authors analyze the "reverse Labouchere" system described in Norman Leigh's (1976) Thirteen against the Bank, which claimed an 800,000-franc win in eight days.
   • Using Corollary 5.1, they establish that the system must converge to a loss of $1/74$ (approx. 1.35%) in the long term.
   • Through Monte Carlo simulation (1 million replications of 8-day sessions) and Poisson approximation, they demonstrate that the reported number of "winning progressions" (27) and the magnitude of the win are statistically impossible under chance. The probability of observing 27 winning progressions is estimated to be less than $1.5 \times 10^{-24}$.
   • The authors conclude that the story is fictional, noting that the reported "profit" ignored the frequent small losses inherent to the system.

Significance and Claims
The paper claims to provide a unified, rigorous framework for analyzing the long-term performance of casino games under realistic betting strategies. Its primary significance lies in:

• Generalization: Moving beyond the restrictive i.i.d. assumption to cover dependent bets, compound structures, and delayed resolutions.
• Strategy Independence: Demonstrating that while a gambler's strategy influences the variance and the path of the profit, it cannot alter the asymptotic limit of the profit-to-bet ratio, which is determined solely by the game's intrinsic House Advantage (provided the strategy does not involve unbounded bet sizes that violate the second-moment conditions).
• Empirical Validation: Providing a mathematical refutation of a famous gambling anecdote, illustrating the practical utility of the theoretical bounds in distinguishing between plausible outcomes and statistical anomalies.

The authors maintain a modest tone, noting that their results apply to "mild conditions" on conditional expectations and that the theory applies equally to individual gamblers or the aggregate table from the casino's perspective. They do not propose new betting systems but rather offer a tool to analyze the inevitable long-term failure of any system in negative-expectation games.