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Uncertainty

Uncertainty. Robin Burke GAM 206. Outline. Quiz (30 min) Uncertainty Lots Dice. Quiz. 30 min Quiz Answers. Uncertainty. Many games are probabilistic roll the dice shuffle the cards Some games are not Chess Tic-Tac-Toe. Certainty vs uncertainty. Certainty

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Uncertainty

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  1. Uncertainty Robin Burke GAM 206

  2. Outline • Quiz (30 min) • Uncertainty • Lots • Dice

  3. Quiz • 30 min Quiz Answers

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

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

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

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

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

  9. 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

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

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

  12. Possibility Space • How many possible outcomes? • TTTT, HTTT, THTT, HHTT, TTHT, HTHT, THHT, HHHT, TTTH, HTTH, THTH, HHTH, TTHH, HTHH, THHH, HHHH • 24 = 16 • How many of each type? • 0 H = 1 = 1/16 • 1 H = 4 = 1/4 • 2 H = 6 = 3/8 • 3 H = 4 = 1/4 • 4 H = 1 = 1/16 • This is why you get an extra throw on 4 or 5 in Senet • why do you get an extra throw for a 1?

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

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

  15. Two dice • Random variable • usually we care about the 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

  16. 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?

  17. 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 • a particular choice in rock-paper-scissors has a 1/3 chance of winning • Chance degree of success • in "The Game of Life", a selected card determines your salary • Chance due to physical limitations • the difficulty of the hand-eye coordination needed to perform an action

  18. 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

  19. Psychology • People are lousy probabilistic reasoners • Reasoning errors • Most people would say that the odds of rolling at least one 1 with two die = 1/6 + 1/6 = 1/3 • (what's the real probability?) • 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."

  20. 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?

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

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

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