Probability / Information Theory Robin Burke GAM 224 Fall 2005

Outline Admin Erratum Design groups "Rules" paper Uncertainty Probability Information theory Signal and noise

Erratum The rules of "8 & out" are not isomorphic to "Thunderstorm" Mr. Allen was correct Consider all players draw no ones for seven turns each player should be 1 turn from being out but each player should have 7*(k-1) points none of them can go out next turn Moral it is hard to find a confusing alternative to counting

Design teams 9 people not on teams Please get teamed up! Next milestone Monday

Rules paper Due 10/10 Analysis paper #1: "Rules" You should be playing your game and taking notes Note you cannot use lab machines to do word processing laptops are OK

Important points Thesis "great game" is not a thesis This is a thesis "Inertial navigation, fixed firing direction and accurate collision detection in Asteroids create an environment in which ship orientation is highly coupled, generating emergent forms of gameplay." No thesis = paper will not be graded Documentation game itself, book, lectures other sources if used Missing or inadequate documentation = paper will not be graded

Rules paper 2 Schemas Emergence Uncertainty Information Theory Information Systems Cybernetics Game Theory Conflict Do not use more than one

Rules paper 3 Outlines suggestions Focus do not catalog every rule, every game object identify those items that contribute to your argument depth over breadth

Rules paper 4 Turn in hardcopy in class 10/10 Turn in to turnitin.com on 10/10 Late policy ½ grade per day submit by

Turnitin.com Class id Password katamari

Uncertainty Many games are probabilistic roll the dice shuffle the cards Some games are not Chess Checkers Dots and Boxes

Certainty vs uncertainty Certainty the condition when the outcome of an action is known completely in advance. Some games operate this way Chess Dots and Boxes But even then uncertainty about who will win otherwise what is the point? Strategically-interesting deterministic games are hard to design

Probability Probability is the study of chance outcomes originated in the study of games Basic idea a random variable a quantity whose value is unknown until it is "sampled"

Random variable 2 We characterize a random variable not by its value but by its "distribution" the set of all values that it might take and the percentage of times that it will take on that value distribution sums to 1 since there must be some outcome Probability the fraction of times that an outcome occurs

Single Die Random variable # of spots on the side facing up Distribution each value 1/6 of the time Idealization

Single die Random variable odd or even number of dots Distribution odd or even 50% Distributions are not always uniform

Two dice Random variable sum of the two die values Distribution 2, 12 = 1/36 3, 11 = 1/18 4, 10, = 1/12 5, 9 = 1/9 6, 8 = 5/36 7 = 1/6 Non-uniform not the same as picking a random # between 2-12 dice games use this fact

Computing probabilities Simplest to count outcomes Dice poker roll five die keep best k, roll 5-k becomes your "hand" Suppose you roll two 1s what are the outcomes when your roll the other 3 again to improve your hand?

Outcomes Each possible combination of outcomes of 3 rolls 6 x 6 x 6 = 216 possible outcomes Questions Probability of 3 of kind or better? Probability of 4 of a kind or better? Die #1 Die #2 Die #3 1, 1, 1 6, 6, 1 6, 6, 6

Basic probability theory Repeated trials add but not in a simple way probability of a coin flip = heads? probability of heads in 3 flips? Outcome sequences multiply probability of 3 flips all heads?

Relevance for Games Many game actions are probabilistic even if their effects are deterministic player may not be able to exercise the controls perfectly every time Asking somebody to do the same uncertain task over increases the overall chance of success to succeed on several uncertain tasks in a row decreases (a lot) the overall chance of success

Role of Chance Chance can enter into the game in various ways Chance generation of resources dealing cards for a game of Bridge rolling dice for a turn in Backgammon Chance of success of an action an attack on an RPG opponent may have a probability of succeeding Chance degree of success the attack may do a variable degree of damage Chance due to physical limitations the difficulty of the hand-eye coordination needed to perform an action

Role of Chance 2 Chance changes the players' choices player must consider what is likely to happen rather than knowing what will happen Chance allows the designer more latitude the game can be made harder or easier by adjusting probabilities Chance preserves outcome uncertainty with reduced strategic input example: Thunderstorm

Random Number Generation Easy in physical games rely on physical shuffling or perturbation basic uncertainty built into the environment "entropy" Not at all simple for the computer no uncertainty in computer operations must rely on algorithms that produce "unpredictable" sequences of numbers an even and uncorrelated probability distribution sometime variation in user input is used to inject noise into the algorithm Randomness also very important for encryption

Psychology People are lousy probabilistic reasoners Reasoning errors Most people would say that the odds of rolling a 1 with two die = 1/6 + 1/6 = 1/3 We overvalue low probability events of high risk or reward Example: Otherwise rational people buy lottery tickets We assume success is more likely after repeated failure Example: "Gotta keep betting. I'm due."

Psychology 2 Why is this? Evolutionary theories Pure chance events are actually fairly rare outside of games Usually there is some human action involved There are ways to avoid being struck by lightning We tend to look for causation in everything Evolutionarily useful habit of trying to make sense of the world Result superstition "lucky hat", etc. We are adapted to treat our observations as a local sample of the whole environment but in a media age, that is not valid How many stories in the newspaper about lottery losers?

Psychology 3 Fallacies may impact game design Players may take risky long-shots more often than expected Players may expect bad luck to be reversed

Information theory Information can be public board position in chess private one's own poker hand unknown / hidden (to players) monsters in the next room

Information Theory There is a relationship between uncertainty and information Information can reduce our uncertainty Example The cards dealt to a player in "Gin Rummy" are private knowledge But as players pick up certain discarded cards from the pile It becomes possible to infer what they are holding

Information Theory 2 Classical Information Theory Shannon Information as a quantity how information can a given communication channel convey? compare radio vs telegraph, for example must abstract away from the meaning of the information only the signifier is communicated the signified is up to the receiver

Information Theory 3 Information as a quantity measured in bits binary choices If you are listening on a channel for a yes or no answer only one bit needs to be conveyed Example LOTR

Information Theory 4 If you need a depiction of an individual's appearance many more bits need to be conveyed Because there are more ways that people can look young / old race eye color / shape hair color / type height dress A message must be chosen from the vocabulary of signifier options book: "information is a measure of one's freedom of choice when selecting a message"

Information Theory 5 This is the connection between information and uncertainty the more uncertainty about something the more possible messages there are

Noise Noise interrupts a communication channel by changing bits in the original message increases the probability that the wrong message will be received Redundancy standard solution for noise more bits than required, or multi-channel

Example 1 Gin Rummy Unknown What cards are in Player X's hand? Many possible answers 15 billion (15,820,024,220) about 34 bits of information Once I look at my 10 cards 1.5 billion (1,471,442,973) about 30 bits

Example 1 cont'd I have two Kings Player X picks up a discarded King of Hearts Discards a Queen of Hearts There are two possible states Player X now has one King <- certain Player X now has two Kings <- very likely Possible hands 118 million (118,030,185) about 27 bits factor of 10 reduction in the uncertainty 3 bits of information in the message

Example 1 cont'd Gin Rummy balances privacy of the cards with messages discarding cards picking up known discards The choice of a discard becomes meaningful because the player knows it will be interpreted as a message When it isn't your turn the game play is still important because the messages are being conveyed interpreting these messages is part of the skill of the player trivial to a computer, but not for us

Example 2 Legend of Zelda: Minish Cap Monsters are not all vulnerable to the same types of weapons 10 different weapons (we'll ignore combinations of weapons) Encounter a new monster which weapon to use? 4 bits of unknown information We could try every weapon but we could get killed

Example 2, cont'd Messages the monster iconography contains messages rocks and metal won't be damaged by the sword flying things are vulnerable to the "Gust Jar" etc. the game design varies the pictorial representations of monsters knowing that these messages are being conveyed learning to interpret these messages is part of the task of the player once mastered, these conventions make the player more capable Often sound and appearance combine a redundant channel for the information

Game Analysis Issues Be cognizant of the status of different types of information in the game public private unknown Analyze the types of messages by which information is communicated to the user How does the player learn to interpret these messages? How are redundant channels used to communicate?

Monday No class Meet with your group

Wednesday Systems of Information Cybernetics Ch. 17 & 18