Artificial Intelligence  
Lecture 05 Randomness and Probability  
Edirlei Soares de Lima  
<edirlei.slima@gmail.com>  
Game AI Model  
Pathfinding  
Steering behaviours  
Finite state machines  
Automated planning  
Behaviour trees  
Randomness  
Sensor systems  
Machine learning  
Randomness in Games  
Game programmers have a special relationship with random  
numbers. They can be used for several tasks:  
Damage calculation;  
Critical hits probability;  
Item drop probability;  
Reward probability;  
Enemy stats;  
Spawning enemies and items;  
Shooting spread zones;  
Decision making;  
Procedural content generation;  
Randomness and Probability  
Although most programming languages include functions to  
generate pseudo-random numbers, there are some situations  
where some control over the random numbers is extremely  
important.  
Gaussian Randomness: normal distribution of random numbers.  
Filtered Randomness: manipulation of random numbers so they  
appear more random to players over short time frames.  
Perlin Noise: consecutive random numbers that are related to each  
other.  
Gaussian Randomness  
Normal distributions (also known as Gaussian distributions)  
are all around us, hiding in the statistics of everyday life.  
Height of Trees  
Height of People  
Gaussian Randomness  
Normal distributions (also known as Gaussian distributions)  
are all around us, hiding in the statistics of everyday life.  
Speed of Runners in a Marathon  
Speed of Cars on a Highway  
Gaussian Randomness  
There is randomness in previous examples, but they are not  
uniformly random.  
Example:  
The chance of a man growing to be 170 cm tall is not the same as the  
chance of him growing to a final height of 150 cm tall or 210 cm tall.  
We see a normal distribution with the height of men centered around  
1
70 cm.  
Gaussian Randomness  
Normal Distribution vs. Uniform Distribution:  
Normal Distribution  
Uniform Distribution  
Gaussian Randomness  
The large majority of distributions in life are closer to a  
normal distribution than a uniform distribution.  
Central Limit Theorem: when several independent random  
variables are added together, the resulting sum will follow a  
normal distribution.  
Example: roll and sum  
three 6-sided dices.  
Gaussian Randomness  
Why do most distributions in life follow a normal distribution?  
Almost everything in the universe has more than one contributing  
factor, and those factors have random aspects associated with them.  
Example: what determines how tall a tree will grow?  
Genes, precipitation, soil quality, air quality, amount of sunlight,  
temperature, exposure to insects, ...  
For an entire forest, each tree experiences varying aspects of each  
quality, depending on where the tree is located.  
Gaussian Randomness  
How Gaussian randomness can be generated?  
Box-Muller Transform (Marsaglia polar method):  
public static float NextGaussian()  
{
float v1, v2, s;  
do{  
v1 = 2.0f * Random.Range(0f, 1f) - 1.0f;  
v2 = 2.0f * Random.Range(0f, 1f) - 1.0f;  
s = v1 * v1 + v2 * v2;  
}while (s >= 1.0f || s == 0f);  
s = Mathf.Sqrt((-2.0f * Mathf.Log(s)) / s);  
return v1 * s;  
}
Gaussian Randomness  
We can change the normal distribution according to a specific  
mean and standard deviation:  
public static float NextGaussian(float mean, float std_dev)  
{
return mean + NextGaussian() * std_dev;  
}
We can also guarantee that values never fall outside the  
limits:  
public static float NextGaussian(float mean, float std_dev,  
float min, float max){  
float v;  
do{  
v = NextGaussian(mean, standard_deviation);  
}while (v < min || v > max);  
return v;  
}
Gaussian Randomness  
Testing the gaussian random numbers:  
void Start () {  
Texture2D texture = new Texture2D(128, 128);  
GetComponent<Renderer>().material.mainTexture = texture;  
for (int x = 0; x < 300; x++) {  
texture.SetPixel((int)NextGaussian(64, 10, 0, 128),  
(int)NextGaussian(64, 10, 0, 128),  
Color.black);  
}
texture.Apply();  
}
Applications of Gaussian Randomness  
Uniform  
Distribution Distribution  
Gaussian  
Gun aiming variation.  
Any aspect of an NPC that may  
vary within a population:  
o Average or max acceleration.  
5
0 Bullets  
1
2
4
00 Bullets  
00 Bullets  
00 Bullets  
o Size, width, height, or mass.  
o Fire or reload rate for firing.  
o Refresh rate or cool-down rate for  
healing or special abilities.  
o Chance of striking a critical hit.  
o Level of intelligence.  
Exercise 1  
1
) Create a random population of 100 characters whose  
height follow a normal distribution in Unity. You can use  
any object to represent the characters, such as cubes or  
cylinders.  
Randomness Test  
Exercise 1: grab a piece of paper and start writing down 0’s  
and 1’s in a random sequence with a 50% chance of each—do  
it until you have a list of 100 numbers.  
Exercise 2: take out a coin and start flipping it, recording the  
sequence of heads and tails as 0s and 1s. Flip it 100 times and  
write the results in the paper.  
Randomness Test  
Exercise 3: compare the two lists you made to a list created by  
a pseudo-random number generator function, with the same  
5
0% chance of either a 0 or a 1. Example:  
0
1
1101100001100001010000001001011110011100111000110  
0101011011111101001011110011111101011111101000011  
What are the differences between the hand-generated list,  
the coin flip list, and the computer generated one?  
Randomness Test  
It’s very likely that the coin flip and computer generated lists  
contain many more long runs of 0’s or 1’s compared to the  
hand-generated list.  
Most people don’t realize that real randomness almost always  
contains these long runs.  
Most people simply don’t believe a fair coin or real randomness will  
produce those long runs of heads or tails.  
Randomness in Games  
Many games include situations where a uniformly distributed  
random number determines something that affects the player,  
either positively or negatively.  
Players have expectations and they believe in “fair probability”.  
Randomness is too random for many uses in games:  
If the player don’t believes in the game randomness, he/she will thing  
that the game is either broken or cheatingall of which are terrible  
qualities to attribute to a game or an AI.  
Randomness in Games  
We have now entered the realm of psychology, and we have  
temporarily left mathematics.  
If the player thinks the game is cheating, then the game effectively is  
cheating despite what is really happening.  
Perception is far more important than reality when it comes to the  
player’s enjoyment of the game.  
Solution?  
Make the numbers slightly less random!  
When generating a random sequence of numbers, if the next number  
will hurt the appearance of randomness, pretend that you never saw it  
and generate a new number.  
Identifying Anomalies  
What makes a sequence of random numbers look less  
random?  
1
. The sequence has a pattern that stands out (e.g. 11001100 or  
11000).  
. The sequence has a long run of the same number (e.g.  
1011111110).  
1
2
0
The goal is to write some rules to identify these anomalies,  
and then throw out the last number that triggers a rule.  
Filtering Binary Randomness  
Rules:  
1
. If the newest value will produce a run of 4 or more equal values,  
then there is a 75% chance to flip the newest value.  
This doesn’t make runs of 4 or more impossible, but progressively much less likely  
(the probability of a run of 4 occurring goes from 1/8 to 1/128).  
2. If the newest value causes a repeating pattern of four values, then  
flip the last value.  
Example: 11001100 becomes 11001101  
3. If the newest value causes a repeating pattern of two values with  
three repetitions each, then flip the last value.  
Example: 111000 becomes 111001  
Filtering Binary Randomness  
Original sequence:  
0
1
1101100001100001010000001001011110011100111000110  
0101011011111101001011110011111101011111101000011  
Filtered sequence (highlighted numbers are flipped):  
0
1
1101100011000101010001001001011100111001110010110  
0101011011101101001011100111011101011101101000110  
Filtering Binary Randomness  
public class BinaryRandom {  
private List<int> generatedNumbers;  
private int maxHistory;  
public BinaryRandom(int historySize){  
maxHistory = historySize;  
generatedNumbers = new List<int>();  
}
public int NextBinary(){  
int value = Random.Range(0, 2);  
if (generatedNumbers.Count > maxHistory)  
generatedNumbers.RemoveAt(0);  
if (FilterValue(value))  
value = FlipValue(value);  
generatedNumbers.Add(value);  
return value;  
}
...  
Filtering Binary Randomness  
...  
private int FlipValue(int value){  
if (value == 1)  
return 0;  
else  
return 1;  
}
private bool FilterValue(int value){  
if (FourRunsBinaryRule(value))  
return true;  
if (FourRepetitionsPatternBinaryRule(value))  
return true;  
if (TwoRepetitionsPatternBinaryRule(value))  
return true;  
return false;  
}
...  
Filtering Binary Randomness  
...  
private bool FourRunsBinaryRule(float value){  
if (generatedNumbers.Count < 3)  
return false;  
for (int i = generatedNumbers.Count - 1;  
i >= generatedNumbers.Count - 3; i--)  
{
if (generatedNumbers[i] != value)  
return false;  
}
if (Random.Range(0, 4) == 0)  
return false;  
return true;  
}
Rule 1: if the newest value will produce a  
run of 4 or more equal values, then there  
is a 75% chance to flip the newest value.  
...  
Filtering Binary Randomness  
...  
private bool FourRepetitionsPatternBinaryRule(float value){  
if (generatedNumbers.Count < 7)  
return false;  
if (generatedNumbers[generatedNumbers.Count - 1] != value)  
return false;  
int count = 0;  
for (int i = generatedNumbers.Count - 2;  
i >= generatedNumbers.Count - 7; i-=2)  
{
if (generatedNumbers[i] == generatedNumbers[i - 1])  
count++;  
}
if (count < 3)  
return false;  
return true;  
Rule 2: if the newest value causes a  
repeating pattern of four values, then  
flip the last value.  
}
.
..  
Filtering Binary Randomness  
...  
private bool TwoRepetitionsPatternBinaryRule(float value){  
if (generatedNumbers.Count < 5)  
return false;  
if ((generatedNumbers[generatedNumbers.Count - 1] != value) ||  
(generatedNumbers[generatedNumbers.Count - 2] != value))  
return false;  
for (int i = generatedNumbers.Count - 3;  
i >= generatedNumbers.Count - 5; i--)  
{
if (generatedNumbers[i] == value)  
return false;  
}
return true;  
}
}
Rule 3: if the newest value causes a repeating  
pattern of two values with three repetitions  
each, then flip the last value.  
Filtering Integer Ranges  
Rules:  
1
2
3
4
. Repeating numbers.  
Example: [7, 7] or [3, 3].  
. Repeating numbers separated by one digit.  
Example: [8, 3, 8] or [6, 2, 6].  
. A counting sequence of 4 that ascends or descends.  
Example: [3, 4, 5, 6].  
. Too many values (4) at the top or bottom of a range within the last  
10 values.  
Example: [6, 8, 7, 9, 8, 6, 9].  
5. Patterns of two numbers that appear in the last 10 values.  
Example: [5, 7, 3, 1, 5, 7].  
6
. Too many (4) of a particular number in the last 10 values.  
Example: [9, 4, 5, 9, 7, 8, 9, 0, 2, 9].  
Filtering Integer Ranges  
Original sequence:  
2
5
2312552222577750677564061448482102435500989388459  
9607889964957780753281574605482138446235103745368  
Filtered sequence (highlighted numbers are thrown out):  
2
5
2312552222577750677564061448482102435500989388459  
9607889964957780753281574605482138446235103745368  
Exercise 2  
2
) Based on the binary filter, create a class to filter integer  
ranges according to the following rules:  
1. Avoid repeating numbers (e.g.: [7, 7] or [3, 3]).  
2
. Avoid repeating numbers separated by one digit (e.g.: [8, 3, 8] or  
[
6, 2, 6].  
. Avoid ascends or descends counting sequences of 4 numbers  
e.g.: [3, 4, 5, 6]).  
. Avoid 4 repetitions of a particular number in the last 10 values  
e.g.: [9, 4, 5, 9, 7, 8, 9, 0, 2, 9]).  
3
4
(
(
Filtering Floating-Point Ranges  
Rules:  
1. Reroll if two consecutive numbers differ by less than 0.02.  
Example: [0.875, 0.856].  
2
. Reroll if three consecutive numbers differ by less than 0.1.  
Example: [0.345, 0.421, 0.387].  
3. Reroll if there is an increasing or decreasing run of 5 values.  
Example: [.342, 0.572, 0.619, 0.783, 0.868].  
4. Reroll if there are too many values (4) at the top or bottom of a  
range within the last 10 values.  
Example: [0.325, 0.198, 0.056, 0.432, 0.119, 0.043].  
Perlin Noise for Game AI  
Perlin noise is a type of gradient noise typically used in  
computer graphics to generate organic textures.  
Perlin noise generates a form of coherent randomness, where  
consecutive random numbers are related to each other.  
This “smooth” nature of randomness don’t generates wild jumps from  
one random number to another, which can be a very desirable trait.  
Perlin Noise for Game AI  
Possible applications of Perlin noise for game AI:  
Movement (direction, speed, acceleration);  
Layered onto animation (adding noise to facial movement or gaze);  
Attention (guard alertness, response time);  
Play style (defensive, offensive);  
Mood (calm, angry, happy, sad, depressed, manic, bored, engaged);  
Perlin Noise in Unity  
Unity has a function to compute 2D Perlin noise:  
float Mathf.PerlinNoise(float x, float y);  
It returns the Perlin noise value between 0.0 and 1.0.  
Although the noise plane is two-dimensional, we can ignore  
one coordinate and sample the noise from just one-dimension.  
Perlin Noise in Unity  
Example: movement direction:  
public class WanderAgent : MonoBehaviour  
{
public float speed = 2;  
public float rotationFactor = 1.2f;  
public float seed = 0.5f;  
void Update ()  
{
transform.forward = new Vector3(Mathf.PerlinNoise(Time.time *  
seed, 0.0f) * rotationFactor,  
transform.forward.y, transform.forward.z);  
transform.position += transform.forward * Time.deltaTime * speed;  
}
}
Further Reading  
Rabin, S., Goldblatt, J., and Silva, F. (2013). Game AI Pro: Collected  
Wisdom of Game AI Professionals. Steven Rabin (ed.), A K Peters/CRC  
Press, ISBN: 978-1466565968.  
Chapter 3: Advanced Randomness Techniques for Game AI