Transcript MATH AND GAME BALANCE

CHAPTER 11
MATH AND GAME BALANCE
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Topics
 The Importance of Math in Game Balance
 Using Spreadsheets to Balance Games
 The Math of Probability
– Jesse Schell's 10 Rules of Probability
 Weighted Distributions
 Randomizer Technologies in Paper Games
 Permutations
 Positive and Negative Feedback Systems
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The Importance of Math in Game Balance
 Balance means different things depending on the
context
– In Multiplayer Games, balance most often means fairness
• Each player should have an equal chance of winning the game
• Most easily accomplished in symmetric games where each player
has the same starting point and abilities
• Balancing an asymmetric game is considerably more difficult
– And requires extensive playtesting
– In Single-Player Games, balance means
• The game is at an appropriate level of difficulty for the player
• The difficulty changes gradually
– If a game has a large jump in difficulty at any point, that point becomes a
place where the game will tend to lose players
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The Importance of Math in Game Balance
"Change a number, and you change
the experience. Period."
– Chris Swain
Game Designer & Professor
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The Importance of Math in Game Balance
 Because numbers have such a profound effect on
games, you must understand some simple math
concepts to balance games well
 In this chapter, you learn about several disparate
aspects of math that are all part of game design and
balance
 This includes:
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–
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Understanding probability
An exploration of different randomizers for paper games
Weighted distribution
Permutations
Positive and negative feedback
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Using Spreadsheets to Balance Games
 Spreadsheet programs are often used by game
designers to explore issues of balance
 The book chapter uses OpenOffice Calc to explore
math and probability
– Excel, Google Docs Spreadsheets, or LibreOffice Calc could
also be used
– But each program is slightly different
– OpenOffice was chosen because it is free & easily acquired
 Spreadsheets are used in game balance because
– They can help you quickly grasp gestalt information
• Which allows you to make decisions based on numbers, not instinct
– Spreadsheet charts can help convince nondesigners of the
validity of game design decisions
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Using Spreadsheets to Balance Games
Read the chapter to learn about using Calc
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The Math of Probability
There are 36 different possible rolls of 2d6
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The Math of Probability
 Writing down every roll like this is enumeration
 It's also possible to determine the number of possible
rolls with math
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6
–
x6
– = 36
possible rolls for Die A
possible rolls for Die B
total possible rolls
 Probability as a mathematical study was started by
Blaise Pascal and Pierre de Fermat
 In The Art of Game Design, Jesse Schell lists 10 rules
of probability that all game designers should know
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Schell's Ten Rules of Probability
 Rule #1 - Fractions = Decimals = Percents
– All three are different ways to represent the same numbers
– The chance of rolling a 1 on 1d20 (a 20-sided die) is
• 1/20 as a Fraction
• 0.05 as a Decimal
• 5% as a Percent
– To convert between them:
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•
•
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Fraction to Decimal - Type the fraction into a calculator
Percent to Decimal - Divide by 100 (e.g., 5% = 5/100 = 0.05)
Decimal to Percent - Multiply by 100 (e.g., (0.05 * 100)% = 5%)
Anything to Fraction - Unfortunately, there's no easy way to do so
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Schell's Ten Rules of Probability
 Rule #2 - Probabilities range from 0 to 1
– According to Rule #1
• 0 to 1 and 0/1 to 1/1 and 0% to 100% are the same
– There can never be less than a 0% chance
– There can never me more than a 100% chance
 Rule #3 - Probability is Sought Outcomes divided by
Possible Outcomes
– Example: Wanting to roll a 6 on 1d6
• Sought outcomes: 1 (the roll of 6)
• Possible outcomes: 6 (the number of sides of the die)
• Probability: 1/6 (≈1.666666 or ≈17%)
– Example: Wanting to draw a spade from a deck of cards
• Sought outcomes: 13 (the number of spades in the deck)
• Possible outcomes: 52 (total number of cards)
• Probability: 13/52 = 1/4 (0.25 or 25%)
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Schell's Ten Rules of Probability
 Rule #4 - Enumeration can solve simple problems
– If you have a low number of possible outcomes, enumeration
can work
• e.g., the 2d6 example on slide 8
– However, 10d6 has 60,466,176 possible rolls!
• This can only be reasonably enumerated by a computer
• Appendix B, "Useful Concepts" contains a program to do this
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Schell's Ten Rules of Probability
 Rule #5 - When sought outcomes are mutually
exclusive, add their probabilities
– Example: Drawing either an Ace or a Face Card from a deck
• There are no cards that are both an Ace and a Face Card
– So these sought outcomes are mutually exclusive
• Sought outcomes:
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–
4
12
Aces
Face Cards
• Possible outcomes: 52 total cards in the deck
• Probability: 4/52 + 12/52 = 16/52 = 4/13 (≈0.3077 or ≈31%)
– Example: Rolling 1, 2, or 3 on 1d6
• Sought outcomes: 3 (1, 2, or 3)
• Possible outcomes: 6 (number of sides of the die)
• Probability: 3/6 = 1/2 (0.5 or 50%)
– If you use OR to describe sought outcomes, you can add
them together
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Schell's Ten Rules of Probability
 Rule #6 - When sought outcomes are not mutually
exclusive, multiply their probabilities
– Example: Drawing a card that is a Face Card and a Spade
• These sought outcomes are NOT mutually exclusive
• Probability for each case:
–
12/52 to draw a Face Card AND 13/52 to draw a Spade
• Total Probability: 12/52 x 13/52
= (12 x 13) / (52 x 52)
=
156 / 2704
=
3 / 52 (≈0.0577 or ≈6%)
– Example: Rolling two sixes on 2d6
• Sought outcomes: 1 on Die A and 1 on Die B
• Possible outcomes: 6 on Die A and 6 on Die B
• Probability: 1/6 x 1/6 = 1/36 (≈0.0278 or ≈3%)
– If you use AND to describe sought outcomes, you multiply
them together
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Schell's Ten Rules of Probability
 Rule #7 - One minus "does" = "doesn't"
– The chance of something happening is one minus the chance
of it NOT happening
– Sometimes it's easier to figure the chance of something not
happening and reverse it
– Example: The chance of rolling at least one 6 on 2d6
• Sought outcomes: 6_6 OR 6_x OR x_6 (x means a non-6)
• These are mutually exclusive (with an OR), so add
• Probability: ( 1/6 x 1/6 ) + ( 1/6 x 5/6 ) + ( 1/6 x 5/6 )
= 1/36 + 5/36 + 5/36 = 11/36
– Example: The chance of Die A rolling a non-six AND Die B
rolling a non-six (the opposite of the previous example)
• These sought outcomes are NOT mutually exclusive, so multiply
• Probability: 5/6 x 5/6 = 25/36
– Therefore, the chance of rolling at least one 6 on 2d6 is
• Probability: 1 - 25/36 = 11/36 (≈0.3055 or ≈30.6%)
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Schell's Ten Rules of Probability
 Rule #8 - The sum of multiple dice is not a linear
distribution
– With multiple dice, the probability of sums is a bell curve
– The more dice, the more extreme the bell curve
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Let's zoom in!
Schell's Ten Rules of Probability
 Rule #8 - The sum of multiple dice is not a linear
distribution With 4d6, you're 146 times more likely
tothe
roll probability
a 14 than a 4of
orsums
a 24 is a bell curve
– With multiple dice,
– The more dice, the more extreme the bell curve
With 10d6, you're about 4.4 million times more
likely to roll a 35 than a 10 or a 60!!!
35: 4,395,456 / 60,466,176 ≈ 7%
52: 23,760 / 60,466,176 ≈ 0.04%
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Schell's Ten Rules of Probability
 Rule #9 - Theoretical vs. practical probability
– Results of real rolls will differ from theoretical predictions
– This happens for several reasons
• Many dice are not perfectly balanced
• Probability is only the likelihood that something will happen, not a
guarantee
– For 2d6, you can guarantee theoretical results by printing a
deck of 36 cards (one 2, six 7s, two 11s, etc.)
• Only shuffle the deck once it's been completely exhausted
• This is an option for Settlers of Catan and was implemented in Tetris
– You can also write a computer program to roll dice randomly
several million times – The Monte Carlo approach
• The results will be similar to the theoretical bell curve but not exact
• This could be used to experimentally determine which properties on a
Monopoly board are most likely to be landed on
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Schell's Ten Rules of Probability
 Rule #10 - Phone a friend
– Nearly all college computer science and math students take
classes in probability
• Try asking one of them if you're stuck on a probability problem
– This is how the study of probability began
• The Chevalier de Méré wanted to know why
– He won when he bet someone he would roll one 6 on 4 rolls of 1d6
– He lost when he bet someone he would roll one 12 on 24 rolls of 2d6
– The Chevalier asked his friend Blaise Pascal for help, and
Pascal began corresponding Pierre de Fermat about it
– Probability of one 6 in 4 rolls of 1d6
•
1 - ( 5/6 x 5/6 x 5/6 x 5/6 )
= 1 - ( 625 / 1,296 )
= ( 1,296 - 625 ) / 1,296
= 671 / 1,296 ( ≈ 0.5177 or ≈ 52% )
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Schell's Ten Rules of Probability
 Rule #10 - Phone a friend
– Probability of one 12 in 24 rolls of 2d6
• Probability of one 12 in a roll of 2d6: 1/36
• Probability of NOT rolling a 12 on a roll of 2d6: 35/36
• Probability of NOT rolling a 12 on 24 rolls of 2d6:
( 35/36 )24 ≈ ( 0.97222 )24 ≈ 0.5086 ≈ 50.8%
• Probability of rolling one 12 on 24 rolls of 2d6:
1 - ( 50.8% ) ≈ 49.2%
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Schell's Ten Rules of Probability
 Schell's 10 Rules:
– Rule #1 - Fractions = Decimals = Percents
– Rule #2 - Probabilities range from 0 to 1
– Rule #3 - Probability is Sought Outcomes divided by Possible
Outcomes
– Rule #4 - Enumeration can solve simple problems
– Rule #5 - When sought outcomes are mutually exclusive (OR),
add their probabilities
– Rule #6 - When sought outcomes are not mutually exclusive
(AND), multiply their probabilities
– Rule #7 - One minus "does" = "doesn't"
– Rule #8 - The sum of multiple dice is not a linear distribution
– Rule #9 - Theoretical vs. practical probability
– Rule #10 - Phone a friend
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Weighted Distributions
 A weighted distribution is one in which one result is
more likely than others
– Common example is 2d6 (7 is 6x more likely than 2 or 12)
– Another possibility is having a non-linear distribution of
numbers on the sides of the die
• Example: The attack die from Small World
– 0 occurs 1/2 of the time
– 1, 2, or 3 each occur 1/6 of the time
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Weighted Distributions
 Exercise: Create a specific weighted Die
– Desired probability:
• 0 occurs 1/2 of the time
• 1 is 3x more likely than 3
• 2 is 2x more likely than 3
– Give it a try…
– Resultant die:
• Luckily, d12 is a common die size
• But dice are limited in the way their distributions can be weighted
– Because only certain numbers of sides are easy to manufacture
– Keep these limitations in mind as we look at randomizer technologies
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Randomizer Technologies in Paper Games
 The most common randomizers in paper games are
– Dice
– Spinners
– Decks of Cards
 Each have particular traits that make them attractive
for different situations
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Randomizer Technologies in Paper Games
 Dice
– A single die has a linear probability distribution
– The more dice you add together, the more the result is biased
toward the average
– Standard die sizes include: 4, 6, 8, 10, 12, and 20-sided
– Commonly available packs of dice for gaming usually include
1d4, 2d6, 1d8, 2d10, 1d12, and 1d20.
– 2d10 are sometimes called Percentile Dice
• One die is used for the 1s place (marked with the numbers from 0–9)
• The other for the 10s place (marked with the multiples of 10 from 00–
90)
• Rolled together, they give an even distribution of the numbers from 00
to 99 (where a roll of 0 and 00 is usually counted as 100%).
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Randomizer Technologies in Paper Games
 Spinners
– All spinners have a rotating element and a still element
• In the image, the gray element is still, and the black rotates
– Spinners are often used for children's games
• Young children can't roll dice with accuracy
• Spinners are more difficult to swallow than dice
– Spinners also provide some distinct advantages over dice
• Spinners can have any number of slots (it's difficult to make a d7)
• Spinners can have any kind of weighted distribution without the limits
of a die(image C below)
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Randomizer Technologies in Paper Games
 Cards
– A standard deck of playing cards includes
• 13 ranks of 4 different suits ( Ranks: A-10, J, Q, & K )
• Suits are usually Clubs, Diamonds, Hearts, & Spades
• Sometimes two Jokers
Vectorized Playing Cards 1.3 ( http://code.google.com/p/vectorized-playing-cards/ ) ©2011 - Chris Aguilar Licensed under LGPL 3
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Randomizer Technologies in Paper Games
 Cards
– When drawing a single card, you have these probabilities
• Chance of drawing a particular single card: 1/52 (0.0192 ≈ 2%)
• Chance of drawing a specific suit: 13/52 = 1/4 (0.25 = 25%)
• Chance of drawing a face card: 12/52 = 3/13 (0.2308 ≈ 23%)
– Custom Card Decks
• Very easy to make!
• Easily reconfigurable!
– When to shuffle?
• Shuffle the deck before each draw to have it act like a die roll
– An equal chance of drawing any single card
• Shuffle the deck after it's exhausted to enforce theoretical probability
– Each card will be seen once before any are seen a second time
– However, this can allow card counting
• For a standard 52-card deck, according to mathematician and
magician Persi Diaconis, seven good riffle shuffles should be sufficient
to create a truly random distribution
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Permutations and Combinations
 Bulls and Cows (Traditional Game)
– The basis for Mastermind (1970 by Mordecai Meirowitz)
– Players take turns trying to guess their opponent's code
– The code is a series of 4 digits (0-9)
• Each digit must be different from the others
– For each guess, the guesser receives a number of bulls and
cows
• Bull: For each number that is in the correct place
• Cow: For each number that is in the code but in the wrong position
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Permutations and Combinations
 Bulls and Cows (Traditional Game)
– Try playing the game with a partner now
– …
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Permutations and Combinations
 Permutations
– In math, a selection of elements where the order matters
– The code in Bulls & Cows is a permutation without repetition
 Permutations With Repeating Elements
– A bit easier to calculate than those without repetition
– To find the number of possible permutations, multiply the
number of choices per slot by the number of slots
– For a selection of four digits (0-9)
• 10 x 10 x 10 x 10 = 10,000 possible permutation
 Permutations Without Repeating Elements
– Subtract 1 from possible choices for each subsequent slot
– For Bulls & Cows
• 10 x 9 x 8 x 7 = 5,040 possible permutations without repetition
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Permutations and Combinations
 Combinations
– In math, a selection of elements where order doesn't matter
– 1234, 1423, 4231, and 2431 are all the same combination
 Example Combination
– How many possible combinations of three scoops of ice
cream are there if there are 8 flavors?
• For each scoop, there are 8 possible flavors
• But, the order of the scoops don't matter, so 123 is the same as 213
– Written 8C3 in math (from 8 possibilities, choose 3)
– Formula: n is the number of possibilities, k the # of choices
• nCk = n! / ( k! * (n-k)! )
• 8C3 = 8! / (3! * (8-3)! )
= 8*7*6*5*4*3*2*1 / ( 3*2*1 * ( 5*4*3*2*1) )
= 8*7*6*5*4*3*2*1 / ( 3*2*1 * ( 5*4*3*2*1) )
= 8*7*6 / 3*2*1 = 336 / 6 = 56 combinations
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Permutations and Combinations
 Combinations
– This can also be solved using Pascal's Triangle
1
1
1
1
1
1
1
2
3
4
1
3
6
1
4
1
5
10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
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Permutations and Combinations
 Combinations
– This can also be solved using Pascal's Triangle
Choices
0
1
Possibilities
2
3
4
5
6
7
8
9
0
1
2
3
1
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
8
9
1
3
6
10
15
21
28
36
1
4
10
20
35
56
84
4
5
1
5 1
15 6
35 21
70 56
126 126
6
7
1
7 1
28 8
84 36
8
9
1
9
1
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Positive and Negative Feedback Systems
 Audio feedback occurs when a microphone is placed
in front of a speaker
– The sound from the speaker keeps being re-amplified
– You may have experienced this in multi-person voice chat
 All games have feedback systems
– Actions and results in one turn affect subsequent turns
 Game feedback can either be positive or negative
– Positive - If someone starts winning, their chances of winning
get better and better
• "Positive" in this case does NOT always mean "good"
– Negative - If someone starts winning the game, negative
feedback slows them down and helps other people catch up
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Positive and Negative Feedback Systems
 Features of Positive Feedback Games
–
–
–
–
–
The winner keeps winning
An early lead often determines the winner
Generally frowned upon in multiplayer games
But, it can keep games short!
Single-player games often have positive feedback to help the
player feel more powerful
 Examples of Positive Feedback Games
– Poker
• Once a player has more chips than everyone else, it's easier for her to
bluff and force others to fold
– Monopoly
• Once a player has more money than everyone else, she will continue
to acquire more property and thereby more rent
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Positive and Negative Feedback Systems
 Features of Negative Feedback Games
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–
–
–
–
The winner is slowed
An early lead often does not determine the winner
The last-place player is given bonuses to help her
Tends to make multiplayer games take longer
Rarely used in single-player games
 Examples of Negative Feedback Games
– Mario Kart
• Players in the worst places get speed boosts
• Players in the worst places "randomly" get much better weapons
– Last place players get items like the Lightning Bolt, which slows down
every other player in the race
– First place players will only get bananas (a defensive weapon), a set of
three bananas, or green shells (the weakest attack)
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Positive and Negative Feedback Systems
 Adding Positive or Negative Feedback to Basketball
– Positive Feedback
• If a team is ahead by 10 points, they get an extra player on the court
– Negative Feedback
• If a team is behind by 10 points, they get an extra player on the court
 Discussion:
– What are some positive or negative feedback mechanisms in
your favorite multiplayer games (or board games)?
– How could you alter the rules of your favorite game to
introduce more positive or negative feedback?
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Chapter 11 – Summary
 Games must be balanced to be enjoyable
– Fair for multiple players
– Of appropriate difficulty for single players
 An understanding of math is essential for game
designers to be able to balance games well
 An understanding of spreadsheets helps as well
– The book has a lot more information about spreadsheet use
– The book also includes information about how a spreadsheet
was used to balance the guns in Chapter 9
 Next Chapter: Puzzle Design
– Puzzles feature prominently in single-player games
– Learning to design them can make you a better game
designer
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