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Long-term behavior of casino games

S. N. Ethier, L. Stefanello

TL;DR

The paper develops a general, martingale-based framework to characterize the asymptotic behavior of the ratio of total returns or profits to total bets in casino games where wagers can depend on past outcomes. By leveraging a Doob decomposition and auxiliary results for ratios of sums, it shows that, under mild moment conditions and with appropriate conditional-expectation bounds, the long-run RTP and HA govern the almost-sure limits, even beyond i.i.d. wagers. It provides explicit convergence results and bounds for simple, compound, and future-dependent games, and demonstrates the practical implications through Leigh's roulette case, showing the reported win is extraordinarily unlikely under the model. The framework unifies a wide class of gambling settings, allowing for time-varying strategies and unresolved bets, while preserving interpretable intrinsics RTP and HA as the limiting quantities.

Abstract

We study the asymptotic behavior of the ratio of total return (or total profit) to total amount bet in a casino game. While the limit is well understood when the sequence of wagers is independent and identically distributed, here we consider the case in which bet sizes vary over time and may depend on past outcomes. We propose a general framework that yields such results under mild conditions on the conditional expectations of bets, returns, and profits. The set-up applies to many casino games (including compound games and those in which wagers are not immediately resolved), expressing the long-term behavior in terms of intrinsic parameters, namely return to player (RTP) and house advantage (HA). As an application, we examine the roulette win documented in Leigh's (1976) Thirteen against the Bank and attempt to quantify the likelihood that the story is true.

Long-term behavior of casino games

TL;DR

The paper develops a general, martingale-based framework to characterize the asymptotic behavior of the ratio of total returns or profits to total bets in casino games where wagers can depend on past outcomes. By leveraging a Doob decomposition and auxiliary results for ratios of sums, it shows that, under mild moment conditions and with appropriate conditional-expectation bounds, the long-run RTP and HA govern the almost-sure limits, even beyond i.i.d. wagers. It provides explicit convergence results and bounds for simple, compound, and future-dependent games, and demonstrates the practical implications through Leigh's roulette case, showing the reported win is extraordinarily unlikely under the model. The framework unifies a wide class of gambling settings, allowing for time-varying strategies and unresolved bets, while preserving interpretable intrinsics RTP and HA as the limiting quantities.

Abstract

We study the asymptotic behavior of the ratio of total return (or total profit) to total amount bet in a casino game. While the limit is well understood when the sequence of wagers is independent and identically distributed, here we consider the case in which bet sizes vary over time and may depend on past outcomes. We propose a general framework that yields such results under mild conditions on the conditional expectations of bets, returns, and profits. The set-up applies to many casino games (including compound games and those in which wagers are not immediately resolved), expressing the long-term behavior in terms of intrinsic parameters, namely return to player (RTP) and house advantage (HA). As an application, we examine the roulette win documented in Leigh's (1976) Thirteen against the Bank and attempt to quantify the likelihood that the story is true.
Paper Structure (9 sections, 8 theorems, 93 equations, 1 figure, 3 tables)

This paper contains 9 sections, 8 theorems, 93 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main result
  4. Simple games with one bet per coup
  5. Simple games with multiple bets per coup
  6. Interlude: The Leigh roulette system
  7. Compound games with one bet per coup
  8. Compound games with multiple bets per coup
  9. Games with future dependence

Key Result

Theorem 2.1

Let $(Y_n)_{n\ge 1}$ be a sequence of random variables adapted to a filtration $(\mathscr{F}_n)_{n\ge0}$ such that Consider the martingale $(S_n)_{n\ge 0}$ with respect to $(\mathscr{F}_n)_{n\ge0}$, where $S_0:=0$ and $S_n:=Y_1+\cdots+Y_n$ for $n\ge1$. Suppose that Then

Figures (1)

  • Figure 1: Histogram for the ratio of total profit to total amount bet (in percentage terms) in eight 360-coup days of the Leigh system, based on one million simulated replications.

Theorems & Definitions (29)

  • Theorem 2.1: Chow
  • Lemma 2.2
  • Remark 2.3
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Corollary 4.1
  • Remark 4.2
  • Remark 4.3
  • ...and 19 more