Exploring Formal Verification in Tabletop Game Design
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Chapter 1: Introduction to Formal Verification in Gaming
In my previous discussion, I examined the application of formal verification in game design, particularly the complexities involved in articulating problems for verification. This follow-up explores various strategies I've implemented to refine game rules and deck configurations.
My journey begins with a focus on representing systems as state machines, tailoring the design based on the properties we aim to verify. One major question I faced was determining the optimal card composition for the deck. The arrangement of cards plays a crucial role in ensuring players feel engaged rather than frustrated or disinterested during gameplay.
Section 1.1: Conceptualizing as a State Machine
The essence of gameplay revolves around transferring points from one stock in the deck to another on the board. By conceptualizing each card value as a distinct stock that holds a specific quantity, I can better visualize the system's dynamics.
Subsection 1.1.1: Initial Design Considerations
Section 1.2: The Naive Approach to Design
A common principle in software development applies here: sometimes, starting with a simple, perhaps inefficient, approach can reveal essential insights. A standard deck of playing cards consists of four suits, each with 13 cards. Given my game involves four categories, this serves as a suitable foundation.
The simplest design would be to balance negative and positive cards evenly. However, this creates a scenario where the total card value remains zero, leading to a stalemate as the game progresses—exactly the opposite of what we desire.
To enhance gameplay, incorporating wildcards or cards that modify previously played values can help, but a robust game design should minimize reliance on these mechanisms.
Chapter 2: Testing and Refining Game Mechanics
Description: This video explains how to apply Turing Machines to improve game design, illustrating effective strategies for problem-solving in game mechanics.
The traditional method for resolving these design dilemmas is through test plays, which can be time-consuming and resource-intensive. Creating a prototype deck and recruiting players for testing presents additional challenges. This is why I turned to formal specification as a way to streamline the process, identifying major flaws before engaging in extensive test rounds.
Section 2.1: Simulations as a Testing Ground
My initial step was to create naive simulations in Python. One such simulation revealed that targeting a score of 250 was excessively high. By simplifying the game to involve a single suit and one player, I could experiment with various card distributions and initial values.
From my findings, I concluded that gameplay should ideally allow a win within five moves when utilizing one suit. Expanding to a more sophisticated simulation that allowed for bot interactions further clarified that increasing the number of suits necessitated a decrease in the average rounds required to win.
Subsection 2.1.1: Generalization and Simplification
All specifications require a level of generalization and simplification. It is vital to strike a balance between being general enough to apply axioms and invariants while still remaining representative of reality.
A recent podcast featuring Kathleen Fisher from DARPA emphasized how domain-specific languages can alter our problem-solving approaches. This inspired me to attempt to express my challenges using my personal domain-specific language, Fault. Although only partially implemented, this endeavor helped me debug and improve my compiler.
The goal of my model is to determine the required number of cards in a deck such that the total points of played cards eventually meet the target score.
Section 2.2: Addressing Reachability Challenges
During a discussion about my design issues, a friend introduced me to the concept of reachability, a term I was previously unfamiliar with.
Reachability, distinct from other verification challenges, can be visualized as a graph. For instance, if there are three cards remaining in the deck with values [-10, 10, 30] and the cumulative points from played cards is 80, reaching a goal of at least 100 points is always feasible.
Modeling reachability is complex, as not all verification languages can express it effectively. However, in SAT/SMT contexts, it resembles sudoku solvers, where each potential state corresponds to a variable and logic rules dictate transitions between nodes.
Description: This video covers essential game design tools that go beyond spreadsheets and flowcharts, providing insights into more effective methodologies for game creation.
In constructing a verifiable model, we monitor all variable states where the total played value exceeds the goal. By asserting that at least one of these states is true, we can evaluate the model. If the SMT solver fails to identify a solution, it confirms that reaching the goal is perpetually achievable.
As the specification evolves, we may encounter scenarios where unlikely outcomes become impossible due to the introduction of wildcards—cards that permit players to alter or eliminate previously played ones. Such mechanics may highlight challenges that are unlikely to occur in practice, as they are not incentivized by the game's objectives.
Ultimately, while formal verification can clarify certain issues, it has limitations in addressing irrational strategies within competitive gameplay dynamics.