Games & Virtual Worlds Series

Chance, Probability & Fairness

Understanding Games: How Video Games & Board Games Work

Michael Filimowicz, PhD

Published in

Understanding Games

11 min readJan 3, 2023

Chance is a concept that refers to the probability of a particular outcome occurring. In games, chance is often used to introduce an element of randomness and to create uncertainty about the outcome of an event. This can make the game more interesting and exciting for players, as they cannot predict with certainty what will happen next.

Chance is often introduced into games through the use of dice, cards, or other randomizing mechanisms. For example, in a board game, a player might roll a die to determine how many spaces they can move on their turn. In a card game, players might draw cards from a shuffled deck to determine the actions they can take.

The benefits of using chance in gameplay include adding an element of unpredictability, which can make the game more challenging and engaging for players. It can also help to level the playing field between players of different skill levels, as chance can sometimes trump strategy. Additionally, chance can add an element of luck or luck to the game, which can be especially appealing to casual players.

There are a number of game mechanics that use chance, including:

Dice rolling: This is a common mechanic in board games, where players roll dice to determine the outcome of an action or event. For example, a player might roll a die to determine how many spaces they can move on the game board.

Card drawing: In card games, players often draw cards from a shuffled deck to determine the actions they can take or the events that will occur in the game.

Randomized events: Some games feature events that are randomly triggered or that have random outcomes. This can include things like random encounters in role-playing games or randomly generated levels in video games.

Probability-based mechanics: Some games use probability to determine the likelihood of certain events occurring. For example, a game might use probability to determine the chances of a player successfully hitting a target.

Chance-based mini-games: Many games feature mini-games or side activities that rely on chance. For example, a game might have a gambling mini-game where players can bet on the outcome of a dice roll or a card draw.

Probabilities

Probability is a measure of the likelihood of a particular event occurring. It is typically expressed as a decimal or fraction between 0 and 1, with 0 indicating that an event will never occur and 1 indicating that it will always occur. For example, the probability of rolling a 6 on a single die is 1/6, or about 0.17.

Odds, on the other hand, are a way of expressing the ratio of the probability of an event occurring to the probability of it not occurring. In gambling, odds are often used to express the payouts for different bets. For example, the odds of rolling a 6 on a single die might be expressed as 5:1, which means that for every 5 times the event does not occur (rolling a number other than 6), it will occur once.

Percentages are another way of expressing probability or odds. To convert a probability expressed as a decimal to a percentage, you can multiply it by 100. For example, the probability of rolling a 6 on a single die (0.17) can be expressed as a percentage by multiplying it by 100, giving us a probability of 17%. To convert odds expressed as a ratio to a percentage, you can divide the probability of the event occurring by the probability of it not occurring and then multiply by 100. For example, the odds of rolling a 6 on a single die (5:1) can be expressed as a percentage by dividing 1 by 5 and multiplying by 100, giving us a probability of 20%.

Deterministic & Non Deterministic Games

In a deterministic game, the outcome of an event or action is determined by the rules of the game and the actions of the players. This means that if the same set of circumstances were to occur multiple times, the outcome would be the same each time. Deterministic games rely on skill and strategy, as players can make decisions that will affect the outcome of the game.

In a nondeterministic game, the outcome of an event or action is influenced by an element of chance or randomness. This means that the same set of circumstances might not always lead to the same outcome. Nondeterministic games often involve some element of luck or random chance, and the outcome of the game may be less predictable as a result.

Examples of deterministic games include chess and checkers, where the outcome of a move is determined by the rules of the game and the position of the pieces on the board. Examples of nondeterministic games include roulette and slot machines, where the outcome is influenced by random chance (e.g. the spin of a wheel or the roll of dice).

Random Numbers & the Bell-Shaped Curve

The bell-shaped curve, also known as the normal distribution curve, is a graphical representation of a set of data that follows a normal distribution. A normal distribution is a statistical distribution in which the majority of data points are clustered around the mean (average) value, with fewer data points as you move further away from the mean. The bell-shaped curve is a useful tool for understanding how random events or processes are distributed over time.

When rolling a pair of dice, a bell curve (normal distribution) can be obtained by analyzing the sum of the outcomes from both dice. There are 36 possible combinations, with sums ranging from 2 to 12. The probability of each sum is calculated by dividing the number of ways to achieve that sum by the total number of combinations (36).

For example, the sum of 7 has the highest probability, with six possible combinations (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), resulting in a probability of 6/36 or 1/6. The sum of 2 or 12 has the lowest probability, with only one possible combination each (1+1 or 6+6), resulting in a probability of 1/36.

Plotting these probabilities on a graph shows a symmetrical, bell-shaped curve, with the peak at the mean sum (7) and the tails representing less likely outcomes (2 and 12).

Here is a table showing the probability distribution for rolling two dice:

Outcome Probability

2 1/36
3 2/36
4 3/36
5 4/36
6 5/36
7 6/36
8 5/36
9 4/36
10 3/36
11 2/36
12 1/36

As you can see, the probability of rolling a 7 is the highest, at 6/36, or about 0.17. The probability of rolling any of the other outcomes is lower, with the probability of rolling a 2 or a 12 being the lowest, at 1/36 or about 0.03.

This probability distribution can be visualized as a graph, with the x-axis representing the possible outcomes (2–12) and the y-axis representing the probability of each outcome occurring. The resulting graph would be a bell-shaped curve, with the most likely outcomes occurring near the center of the curve and the less likely outcomes occurring towards the edges.

Pseudo-Random Numbers

A seed is a starting value that is used to generate a sequence of pseudo-random numbers. In a game, a seed can be used to create a series of random events or outcomes that are based on a predictable sequence. This can be useful in a number of ways:

Testing and debugging: By using a seed to generate a set of random numbers, developers can test and debug their game more easily, as they can recreate the same set of random events over and over again.

Multiplayer: In multiplayer games, seeds can be used to create a shared, predictable sequence of random events, which can help to ensure that the game is fair and balanced for all players.

Saving and loading: By using seeds to generate random events in a game, players can save and load their progress without losing the randomness of the game. This can help to make the game more replayable, as players can experience a different sequence of events each time they play.

Predictability: Using seeds to generate pseudo-random numbers can make the game more predictable and strategic, as players can anticipate the likelihood of certain events occurring. This can be especially useful in games that rely heavily on randomness, such as card games.

Uniform & Non-Uniform Distributions

A uniform distribution is a type of probability distribution in which all possible outcomes are equally likely to occur. This means that if you were to generate a set of random numbers using a uniform distribution, each number would have an equal probability of being generated. A uniform distribution can be obtained by rolling just one die, for example, because each number has the same chance of resulting in the throw as any other number.

A nonuniform distribution, on the other hand, is a type of probability distribution in which different outcomes have different probabilities of occurring. This means that some outcomes are more likely to occur than others.

In a game that uses pseudo-random numbers, the distribution of those numbers can be either uniform or nonuniform, depending on the algorithms and techniques used to generate them. For example, if a game uses a uniform distribution to generate random numbers, it means that all possible numbers are equally likely to be generated. On the other hand, if a game uses a nonuniform distribution, it means that some numbers are more likely to be generated than others.

A game might use a nonuniform distribution to create a more realistic or varied experience, while a game that relies on strategy and skill might use a uniform distribution to create a more predictable and balanced game.

Fairness

In game development, the concept of fairness refers to the idea that the game should be balanced and unbiased, with all players having an equal chance of success regardless of their skill level or other factors. A fair game is one in which all players have the same chances of winning and losing, and where the outcome of the game is not determined by random chance alone.

There are a number of ways that game developers can strive for fairness in their games, including:

Balancing gameplay: This involves adjusting the game mechanics and rules to ensure that all players have an equal chance of winning. This can include adjusting the power level of different characters or items, or adjusting the probability of certain events occurring.

Randomization: In games that use randomness or chance, developers can use techniques such as seeding and uniform distribution to ensure that the randomness is fair and unbiased.

Testing and debugging: Developers can test and debug their games to ensure that there are no bugs or exploits that give some players an unfair advantage.

Fairness ensures that the game is enjoyable and engaging for all players.

Rabin’s model of fairness is a theoretical framework for understanding how people perceive fairness in social situations, including games. According to this model, people’s perceptions of fairness are influenced by three factors:

Equity: This refers to the idea that people should be treated equally and receive benefits or burdens in proportion to their contributions or actions.

Reciprocity: This refers to the idea that people should be treated in a way that is consistent with how they have treated others.

Efficiency: This refers to the idea that resources should be allocated in a way that maximizes overall welfare or utility.

According to Rabin’s model, people’s perceptions of fairness are determined by the extent to which a situation or outcome meets these three criteria. For example, in a game, players might perceive the game as fair if they feel that they are being treated equally, if they feel that the game is rewarding them in proportion to their efforts, and if they feel that the game is efficient and not wasting resources.

Rabin’s model of fairness has been widely studied and has had a significant impact on research on fairness and social norms. It has also been applied to a wide range of practical situations, including the design of economic systems, the resolution of disputes, and the analysis of social interactions. It provides a useful framework for understanding how people perceive fairness in different situations, including games.

Bibliography & Further Reading

A Game Design Vocabulary: Exploring the Foundational Principles Behind Good Game Design by Anna Anthropy and Naomi Clark
A Theory of Fun for Game Design by Raph Koster
Advanced Game Design: A Systems Approach by Michael Sellers
An Introduction to Game Studies by Frans Mayra
Basics of Game Design by Michael Moore
Blood, Sweat, and Pixels: The Triumphant, Turbulent Stories Behind How Video Games Are Made Blood, Sweat, and Pixels: The Triumphant, Turbulent Stories Behind How Video Games Are Made by Jason Schreier
Board Game Design Advice: From the Best in the World vol 1 by Gabe Barrett
Building Blocks of Tabletop Game Design: an Encyclopedia Of Mechanisms by Geoffrey Engelstein and Isaac Shalev
Character Development and Storytelling for Games by Lee Sheldon
Chris Crawford on Game Design by Chris Crawford
Clockwork Game Design by Keith Burgun
Elements of Game Design by Robert Zubek
Fundamentals of Game Design by Ernest Adams
Fundamentals of Puzzle and Casual Game Design by Ernest Adams
Game Design Foundations by Brenda Romero
Game Design Workshop by Tracy Fullerton
Game Mechanics: Advanced Game Design by Ernest Adams and Joris Dormans
Game Writing: Narrative Skills for Videogames edited by Chris Bateman
Games, Design and Play: A detailed approach to iterative game design by Colleen Macklin and John Sharp
Introduction to Game Systems Design by Dax Gazaway
Kobold Guide to Board Game Design by Mike Selinker, David Howell, et al
Kobold’s Guide to Worldbuilding edited by Janna Silverstein
Level Up! The Guide to Great Video Game Design, 2nd Edition by Scott Rogers
Narrating Space / Spatializing Narrative: Where Narrative Theory and Geography Meet by Marie-Laure Ryan, Kenneth Foote, et al.
Narrative Theory: A Critical Introduction by Kent Puckett
Narrative Theory: Core Concepts and Critical Debates by David Herman, James Phelan, et al.
Narratology: Introduction to the Theory of Narrative, Fourth Edition by Mieke Bal
Practical Game Design by Adam Kramarzewski and Ennio De Nucci
Procedural Storytelling in Game Design by Tanya X. Short and Tarn Adams
Professional Techniques for Video Game Writing by Wendy Despain
Rules of Play by Salen and Zimmerman
Storyworlds Across Media: Toward a Media-Conscious Narratology (Frontiers of Narrative) by Marie-Laure Ryan, Jan-Noël Thon, et al
Tabletop Game Design for Video Game Designers by Ethan Ham
The Art of Game Design, 3rd Edition by Jesse Schell
The Board Game Designer’s Guide: The Easy 4 Step Process to Create Amazing Games That People Can’t Stop Playing by Joe Slack
The Cambridge Introduction to Narrative by H. Porter Abbott
The Grasshopper, by Bernard Suits
The Routledge Companion to Video Game Studies by Bernard Perron and Mark J.P. Wolf
The Routledge Encyclopedia of Narrative Theory by David Herman
The Ultimate Guide to Video Game Writing and Design by Flint Dille & John Zuur Platten
Unboxed: Board Game Experience and Design by Gordon Calleja
Video Game Storytelling: What Every Developer Needs to Know about Narrative Techniques by Evan Skolnick
Writing for Video Game Genres: From FPS to RPG edited by Wendy Despain
Writing for Video Games by Steve Ince
100 Principles of Game Design by DESPAIN