Download Probability in Games

Complexity and Emergence in Games
(Ch. 14 & 15)
Seven Schemas
• Schema: Conceptual framework concentrating on one
aspect of game design
• Schemas:
– Games as Emergent Systems
– Games as Systems of Uncertainty
– Games as Information Theory Systems
– Games as Systems of Information
– Games as Cybernetic Systems
– Games as Game Theory Systems
– Games as Systems of Conflict
Have You Ever Been Surprised in a
Game?
• Was it scripted?
• Was it a result of Game Design?
• Could a game developer be surprised
from his own game?
Systems and Complexity
(this is NOT
computational
complexity)
• Reminder:
– What is a system?
– Why can game be viewed as systems?
• System’s Complexity: measure of number of ways in which parts
interact within a system:
– Elevator example: simple algorithm used to control one elevator,
when used to control multiple elevators it fails
• Four categories of system’s complexity:
– Fixed
– Periodic
– Complex
– Chaotic
Complexity and Meaningful Play
Ways parts
interact within
a system
Discernable and
integrated
actions
• Are these two related?
– What happens if there is no interaction between play
elements?
• Example : Grid game
– Is it meaningful?
– Lets add complexity: relations between play elements
• Does it achieves meaningful play?
Emergence
• Behavior of the overall system is greater than its parts
• Example: simple operational rules yet unpredictable
results are attained
• Chess: last century versus current century
• Backgammon machine learning example
• Warcraft
• Bluffing
• Obvious example: complex operational rules
• However: complex rules doesn’t make game harder.
Example? –Chess vs Go
Classes of Object Interactions
• Coupled: objects are linked or affect with one another
recursively
– Non-gaming Example: ant colony
– Gaming example?
• Context-dependent: changes are not the same over time
– Non-gaming example: behaviors of the colony under
attack versus harvesting
– Gaming example?
• Emergence in a system occurs when interactions are
coupled and context-dependent
• To achieve emergent behavior in a game: Tuning (or
iterative design) is needed
However: Be careful!
• Example of bad “emergence” in
games?
Exploit 1, exploit 2, exploit 3
• Countering that bad “emergence” can
lead to trouble for game developers
•
Which again reinforces the idea of
iterative design
Let us plan for Wednesday
• Bring laptop/ipad/… to run your game
• Lets discuss Locations
• All students are encouraged to try their classmates’
games
– If you are not showing the game at any point during
the session, play someone else’s and give them
feedback
• In addition, students who are not creating game need
hand to me a filled form at the end of class
Uncertainty and
Games (Ch. 15)
Uncertainty
• If a game outcome is certain can it exhibit meaningful
play?
• Two kinds of uncertainty:
– Macro-level (overall game)
– Micro-level (individual player’s actions)
Uncertainty and Categories of Games
According to AI
For which of the following categories, label with “Yes” those
for which games can have uncertainty
Chance
Deterministic
Perfect
information
Yes
Imperfect
information
Yes
Yes
Yes
Lesson: you don’t have to “rolling a dice” to achieve
uncertainty in a game
Feeling of Randomness
• We can even achieve a feeling of randomness in a
deterministic perfect information game
– Example?
• Danger: designing a chaotic game
– Example?
Probability in Games
• Probability: a mathematical formalization of uncertainty
• Examples of probabilities in games:
 chance to hit
 amount of damage dealt
 Next shape in Tetris
 Initial location in a multiplayer RTS game
 Loot in an MMO
…
• In a game like Chutes and Ladders, players do not make
decisions (probabilities – a random number generator is
deciding-), why is it “fun”?
Probability
•Example. Suppose that you are in a TV show and you
have already earned 1’000.000 so far. Now, the host
propose you a gamble: he will flip a coin if the coin comes
up heads you will earn 3’000.000. But if it comes up tails
you will loose the 1’000.000. What do you decide?
•We know a degree of belief
•Probability theory allows the analysis of decisions
based on the degree of belief
Probability
• Assigns a number between 0 and 1 to events
•The closer an event is to 1, the more likely we believe it
will occur
•The closer an event is to 0, the less likely we believe it
will occur
•Suppose that I flip a “totally unfair” coin (always come heads):
what is the probability that it will come heads:
1
•Suppose that I flip a “fair” coin:
what is the probability that it will come heads: 0.5
Probability Distribution
•The events E1, E2, …, Ek must meet the following
conditions:
•One always occur
•No two can occur at the same time
•The probabilities p1, …, pk are numbers associated with
these events, such that 0  pi  1 and p1 + … + pk = 1
A probability distribution assigns probabilities to events
such that the two properties above holds
Example (Probability Distribution)
Example:
•1000 tickets are sold at a value of $1 each
•100 are selected. The first 90 win $5, the next 9 win
$10, and the last will win $100
•In the example:
• E1 = “holding a $5 ticket”
• E2 = “holding a $10 ticket”
• E3 = “holding a $100 ticket”
• E4 = “holding a losing ticket”
•The probabilities are:
• p1 = .09
• p2 = .009
• p3 = .001
• p4 = .9
•Note that the probabilities add to 1
•Could we add E5 = “holding a winning ticket”?
No! because E5 occurs at the same time as E1, E2 and E3
Probability in Games (II)
• Let us build a probability distribution for the following
game situations (so we have to list the events and their
probabilities):
 chance to hit: statistics
 Next shape in Tetris
 Initial location in a multiplayer RTS game
 Loot in an RPG
• We can use probabilities to building “smart” NPCs:
 Should NPC attack another NPC/avatar?
 “AI” has control of 5 NPC and has to pick one to
fight the NPC/avatar, which one should it choose?
Expected Utility (I)
• We are given a probability distribution:
 The events E1, E2, …, Ek
 The probabilities p1, …, pk associated with these events
 We have the value of those events: U(E1), U(E2), …, U(Ek)
• The Expected Utility (EU):
 EU = p1 * U(E1) + … + pk * U(Ek)
• Examples:
 EU for the fair coin and I bet $10
 EU for the lottery example
Expected Utility: Example
• Coming back to the example:
Suppose that you are in a TV show and you have already
earned 1’000.000 so far. Now, the host propose you a
gamble: he will flip a coin if the coin comes up heads you
will earn 3’000.000. But if it comes up tails you will loose
the 1’000.000. What do you decide?
•
Answer to that depends on:
– the probability of winning $0 or $ 3’000.000
– How much money you currently have
– Expected utility: a measure of how much I would gain
from taking an action
Expected Utility in Games
• Let us design the following game situation by thinking
about the expected utility:
 Player will fight “big bad monster”. What is a
potential probability distribution modeling this and
what would be potential value for the player?
• We can use expected utilities to building even “smarter”
NPCs:
 Should “AI” attack the player?
Warnings about Probabilities (1)
• A random function assigns to k events: E1, E2, …, Ek, the
same probability: 1/k (called “uniform distribution”)
 Example: rolling a dice has 6 events (one for each face)
with probability of each occurring been 1/6
• Problem: Computers cannot make a perfectly random function
 It is “random” in the sense that you cannot predict the
outcome in advance
 But it is not a uniform distribution
 This is due to the fact that computers are intrinsically
deterministic
 But error is too small to matter
Warnings about Probabilities (2)
• People do not always follow the rules of probability:
Experiment with people
Choice was given between A and B and then
between C and D:
A: 80% chance of $4000 C: 20% chance of $4000
B: 100% chance of $3000 D: 25% chance of $3000
Warnings about Probabilities (2)
• Majority choose B over A and C over D
This turns out to be mathematically inconsistent with the
expected utility:
 U(3000) > 0.8U(4000) and
 0.2U(4000) > 0.25U(3000)
 There are no possible values for U(4000) and
U(3000) that will satisfy these two inequalities
•Book discusses other miss-conceptions
Final Thought
• Probability is not required to get a feeling of randomness
in games
• But well laid-out element of chance will result in
meaningful choices
– Or build “smart” NPCs that will enhance game value
• Example: Player needs to decide whether to attack a
monster or not based
 This decision is based on expected utility
 A “feeling” of how success will it be
 Sometimes players can get quite formal
 How worth would it be if successful
 Careful use of macro- and micro-level uncertainty can
result in “epic” game experience