Uncertainty Robin Burke GAM 206

Outline o Quiz (30 min) o Uncertainty o Lots o Dice

Quiz o 30 min Quiz Answers

Uncertainty o Many games are probabilistic o roll the dice o shuffle the cards o Some games are not o Chess o Tic-Tac-Toe

Certainty vs uncertainty o Certainty o the condition when the outcome of an action is known completely in advance. o But even then o uncertainty about who will win o otherwise what is the point? o Deterministic o no chance element involved o Strategically-interesting deterministic games are hard to design o examples?

Race games o Race games o characterized by the goal o get pieces to the end position first o Almost always involve chance o one known exception o Hare and Tortoise (1974) o Distinguished by o the topology of the track o the number of pieces

Taxonomy o Cruciform games o Nyout (Korean) o Pachisi o Parcheesi o Sorry o Tables games o Royal Game of Ur (Mesopotamia) o Game of Twenty (Egypt) o Senet (Egypt) o Liubo (China) o Nard (Persia)

Probability o Probability is the study of chance outcomes o originated in the study of games o Blaise Pascal (1654) o Basic idea o (modern conception) o a random variable o a quantity whose value is unknown until it is "sampled"

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

Lots o Coin flips o Random variable o heads? o true or false o Distribution o true o false o each value ½ of the time

Multiple Lots o More possible outcomes o 0..4 heads o Probabilities? o Not Uniform

Possibility Space o How many possible outcomes? o TTTT, HTTT, THTT, HHTT, TTHT, HTHT, THHT, HHHT, TTTH, HTTH, THTH, HHTH, TTHH, HTHH, THHH, HHHH o 2 4 = 16 o How many of each type? o 0 H = 1 = 1/16 o 1 H = 4 = 1/4 o 2 H = 6 = 3/8 o 3 H = 4 = 1/4 o 4 H = 1 = 1/16 o This is why you get an extra throw on 4 or 5 in Senet o why do you get an extra throw for a 1?

Single die o Random variable o odd or even number of dots o Distribution o odd? o 50% o The same as binary lots

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

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

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

Role of Chance o Chance can enter into the game in various ways o Chance generation of resources o dealing cards for a game of Bridge o rolling dice for a turn in Backgammon o Chance of success of an action o a particular choice in rock-paper-scissors has a 1/3 chance of winning o Chance degree of success o in "The Game of Life", a selected card determines your salary o Chance due to physical limitations o the difficulty of the hand-eye coordination needed to perform an action

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

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

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

Analysis o Structure o Less certainty of outcome o a good player can have bad luck o Different strategic choices o players can make risk / return tradeoffs o Experience o Generation / revelation of chance elements o becomes an activity o moments of dramatic tension

Wednesday o New Unit o "The Checkered Game of Life" o USA, 1860 o Readings o 10/11: Context o Chapter from The Enduring Vision" o 10/16: Game play o Chapter from Huck's Raft o 10/18: Discussion o 2 nd Chapter from Huck's Raft