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The Theory of Games

Games of Chance. The Theory of Games. Today’s question. Given a fixed budget, how should we play the machines?. Some basic probability. What does ‘probability’ mean? Relative frequency of occurrence of an event. v. Long term. The Monty Hall Puzzle. The Monty Hall Puzzle.

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The Theory of Games

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  1. Games of Chance The Theory of Games

  2. Today’s question Given a fixed budget, how should we play the machines?

  3. Some basic probability • What does ‘probability’ mean? • Relative frequency of occurrence of an event v Long term

  4. The Monty Hall Puzzle

  5. The Monty Hall Puzzle

  6. More probability Machine 1 Machine 2

  7. More probability Machine 1 Machine 2 Costs $1 to play Pays you $1001 with probability 1/50 Costs $5 to play Pays you $105 with probability 1/5 Which one is the better machine to play?

  8. More probability Machine 1 Machine 2 Average payoff = ~$18 Average payoff = $16 So always play machine 2!

  9. More probability • Probability • ‘How likely am I to hit the jackpot?’ • Expected value • `How much on average will I make playing this machine?’

  10. Multi-armed bandits

  11. Multi-armed bandits • Two competing forces • ‘Exploration’ – try out a new machine • ‘Exploitation’ – stick with the reliable machine • Examples?

  12. Multi-armed bandits

  13. Recap

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