Long-term behavior of casino games

TL;DR

This paper analyzes the long-term ratio of profit to bets in casino games, considering variable and dependent wager sizes, and provides a general framework applicable to various game types.

Contribution

It introduces a new framework for understanding long-term casino game behavior with dependent and varying wagers, extending beyond i.i.d. assumptions.

Findings

Framework applies to many casino games including compound and delayed resolution games.

Long-term behavior characterized by RTP and house advantage parameters.

Quantitative analysis of roulette win case from Leigh's study.

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.