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?