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Games

Games. Game 1 Each pair picks exactly one whole number The lowest unique positive integer wins Game 2: Nim Example: Player A picks 5, Player B picks 6, Player A picks 13, Player B picks 19, etc. Game 3 Each pair picks exactly one number The number closest to half the average wins. LUPI.

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Games

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  1. Games • Game 1 • Each pair picks exactly one whole number • The lowest unique positive integer wins • Game 2: Nim • Example: Player A picks 5, Player B picks 6, Player A picks 13, Player B picks 19, etc. • Game 3 • Each pair picks exactly one number • The number closest to half the average wins

  2. LUPI • Lowest unique positive integer game • The winner is the person that meets the following criteria • Exactly one person picked the number • There is no smaller number in which exactly one person picked that number

  3. LUPI • Example with 20 participants • 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 8, 10, 50, 99, 100 • 0 does not win (three people guessed this) • 1 does not win (six people guessed this) • 2 does not win (two people guessed this) • 3 does not win (two people guessed this) • 4 wins (exactly one person guessed this)

  4. LUPI • Analysis about NE is too complicated for this class • We will not analyze this game, but many of you probably tried to guess the number that others would choose

  5. Nim • Recall rules • Each person must add a whole number from 1 to 10 when it is her/his turn • The person that hits the winning number wins

  6. Nim • If each person plays optimally, the winning number determines who wins • How do we do this? • A method called backward induction • Suppose, for example, we play to 11

  7. Nim to 11 • The first person picks a number between 1-10 • The second person picks 11  winner

  8. Nim to 22 • The first person picks a number between 1-10 • Suppose I act second • How should I act? • If I pick 11, then I know I can win, because I can repeat the same set of steps to guarantee victory • Example  3, 11, 19, 22

  9. Nim, working backwards • Based on the previous logic, if I can pick numbers that are multiples of 11 less than the winning number, I can guarantee victory • This is what I will call the “path to victory”

  10. Nim, working backwards • Examples of paths to victory • 99  88  77  66  55  44  33  22  11 • 100  89  78  67  56  45  34  23  12  1 • In the first game, the first person to act cannot guarantee victory if the other player knows the path to victory • In the second game, the first player can guarantee victory by choosing 1 and then following the path to victory

  11. Pick half the average • Rules: • Each person picks a number from 0 to 100 • The person that picks the number closest to half of the average wins • In case of a tie, the winners split the prize

  12. Pick half the average • If you assume that each player picks a number randomly between 0 and 100, then I know the average is 50, and I should pick 25 • However, it would be irrational for anyone to pick a number over 50, since it cannot win  Should I pick a number over 25?

  13. Pick half the average • I can repeat this thinking an infinite number of times to reach the NE • Everybody should pick 0 • How many people picked… • 0? • A number over 50?

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