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Formal Modelling and Analysis of Slot Machines

Jan Friso Groote, Sander van Heesch, Matthias Volk

TL;DR

This work addresses the complex problem of determining and understanding RTP for slot machines when players can influence outcomes. It introduces a formal framework based on probabilistic process specifications (via mCRL2) and a quantitative mu-calculus to automatically compute numeric properties such as RTP and to derive optimal player strategies. The approach is demonstrated on simple slot machines and a real machine (Top Spinner from Errèl Industries), yielding precise RTP values (e.g., about $94\%$ for the five- and ten-play-lines games) and enabling extraction of optimal strategies through model checking and strategy synthesis. The results validate the feasibility and usefulness of formal modelling for unambiguous communication of machine behavior to regulators and stakeholders, and show how this method can reveal the impact of different hold-button rules on long-run returns. The work also highlights discrepancies with online RTP claims, the importance of automated, exhaustive analysis over simulation, and opportunities for future extensions to handle larger horizons and more complex rules.

Abstract

Slot machines can have fairly complex behaviour. Determining the RTP (return to player) can be involved, especially when a player has an influence on the course of the game. In this paper we model the behaviour of slot machines using probabilistic process specifications where the intervention of players is modelled using non-determinism. The RTP is formulated as a quantitative modal formula which can be evaluated fully automatically on the behavioural specifications of these slot machines. We apply the method on an actual slot machine provided by the company Errèl Industries B.V. The most useful contribution of this paper is that we show how to describe the behaviour of slot machines both concisely and unequivocally. Using quantitative modal logics there is an extra bonus, as we can quite easily provide valuable insights by a.o. computing the exact RTP and obtaining the optimal player strategies.

Formal Modelling and Analysis of Slot Machines

TL;DR

This work addresses the complex problem of determining and understanding RTP for slot machines when players can influence outcomes. It introduces a formal framework based on probabilistic process specifications (via mCRL2) and a quantitative mu-calculus to automatically compute numeric properties such as RTP and to derive optimal player strategies. The approach is demonstrated on simple slot machines and a real machine (Top Spinner from Errèl Industries), yielding precise RTP values (e.g., about for the five- and ten-play-lines games) and enabling extraction of optimal strategies through model checking and strategy synthesis. The results validate the feasibility and usefulness of formal modelling for unambiguous communication of machine behavior to regulators and stakeholders, and show how this method can reveal the impact of different hold-button rules on long-run returns. The work also highlights discrepancies with online RTP claims, the importance of automated, exhaustive analysis over simulation, and opportunities for future extensions to handle larger horizons and more complex rules.

Abstract

Slot machines can have fairly complex behaviour. Determining the RTP (return to player) can be involved, especially when a player has an influence on the course of the game. In this paper we model the behaviour of slot machines using probabilistic process specifications where the intervention of players is modelled using non-determinism. The RTP is formulated as a quantitative modal formula which can be evaluated fully automatically on the behavioural specifications of these slot machines. We apply the method on an actual slot machine provided by the company Errèl Industries B.V. The most useful contribution of this paper is that we show how to describe the behaviour of slot machines both concisely and unequivocally. Using quantitative modal logics there is an extra bonus, as we can quite easily provide valuable insights by a.o. computing the exact RTP and obtaining the optimal player strategies.
Paper Structure (32 sections, 28 equations, 8 figures, 2 tables)

This paper contains 32 sections, 28 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Process specifications
  3. Probabilistic process specifications
  4. Modeling slot machines
  5. Related work
  6. Outline
  7. Acknowledgements.
  8. The specification formalisms
  9. Probabilistic process expressions
  10. Data types
  11. Behaviour
  12. Quantitative modal logic
  13. Simple slot machines
  14. A one column slot machine
  15. A three column slot machine
  16. ...and 17 more sections

Figures (8)

  • Figure 1: The Top Spinner slot machine by Errèl Industries B.V.
  • Figure 2: The mCRL2 data types for the five-play-lines and ten-play-lines games
  • Figure 3: The mCRL2 specification of the five-play-lines game
  • Figure 4: The mCRL2 behaviour of the ten-play-lines game
  • Figure 5: The mCRL2 data description of the reel game
  • ...and 3 more figures