Long-term behavior of casino games

Session Title

Gambling Mathematics: Casino Game Theory

Presentation Type

Paper Presentation

Start Date

26-5-2026 12:00 AM

Abstract

One of the central questions in gambling mathematics is how total return (or total profit) compares, in the long term, with the total amount bet. This ratio not only measures how favorable or unfavorable a game is, but also quantifies the extent to which it favors one side or the other (typically, the casino). Although this problem has been widely studied through fundamental parameters such as return to player (RTP) and house advantage (HA), its analysis usually relies on strict and unrealistic assumptions. In particular, wagers are typically assumed to be independent and identically distributed, effectively modeling repetitive and nonadaptive play. In this presentation, we introduce a general framework for studying long-term return and profit under very mild assumptions on the sequence of wagers. We show that players may adapt their betting behavior arbitrarily over time, yet no betting strategy can escape the constraints imposed by the game’s intrinsic parameters in the long term. The analysis relies on martingale methods, in particular strong laws of large numbers for martingales. The set-up applies to a broad class of games, including compound games and those with delayed resolution. Representative examples include roulette, slot machines, and craps.

Author Bios

Lorenzo Stefanello is a mathematician with broad research interests. He earned a PhD in Mathematics from the University of Pisa, where his work focused on topics at the intersection of abstract algebra and number theory. After the PhD, he shifted toward gambling mathematics, pursuing both theoretical research as an independent researcher and applied work as a game mathematician in the gaming industry.

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May 26th, 12:00 AM

Long-term behavior of casino games

One of the central questions in gambling mathematics is how total return (or total profit) compares, in the long term, with the total amount bet. This ratio not only measures how favorable or unfavorable a game is, but also quantifies the extent to which it favors one side or the other (typically, the casino). Although this problem has been widely studied through fundamental parameters such as return to player (RTP) and house advantage (HA), its analysis usually relies on strict and unrealistic assumptions. In particular, wagers are typically assumed to be independent and identically distributed, effectively modeling repetitive and nonadaptive play. In this presentation, we introduce a general framework for studying long-term return and profit under very mild assumptions on the sequence of wagers. We show that players may adapt their betting behavior arbitrarily over time, yet no betting strategy can escape the constraints imposed by the game’s intrinsic parameters in the long term. The analysis relies on martingale methods, in particular strong laws of large numbers for martingales. The set-up applies to a broad class of games, including compound games and those with delayed resolution. Representative examples include roulette, slot machines, and craps.