Time for playing gamesGamesLUPISlide 4Slide 5NimSlide 7Nim to 11Nim to 22Nim, working backwardsSlide 11Pick half the averageSlide 13Slide 14Time for playing gamesForm pairsYou will get a sheet of paper to play games withYou will have 12 minutes to play the games and turn them inGamesGame 1Each pair picks exactly one whole numberThe lowest unique positive integer winsGame 2Player A goes first, picks a whole number from 1 to 10Player B goes next, adds to A’s choice by a number from 1 to 10Players A and B alternate until one person reaches the winning numberExample: Player A picks 5, Player B picks 6, Player A picks 13, Player B picks 19, etc.Game 3Each pair picks exactly one numberThe number closest to half the average winsLUPILowest unique positive integer gameThe winner is the person that meets the following criteriaExactly one person picked the numberThere is no smaller number in which exactly one person picked that numberLUPIExample with 20 participants0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 8, 10, 50, 99, 1000 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)LUPINE is complicated Many of you probably tried to figure out what everyone else in the class guessedExample: If you believe that everyone else will pick 0, I should pick 1Example: If you believe that everyone else will pick 0 or 1 with probability ½, then I should pick 2NimRecall rulesEach person must add a whole number from 1 to 10 when it is her/his turnThe person that hits the winning number winsNimIf each person plays optimally, the winning number determines who winsHow do we do this?A method called backward inductionSuppose, for example, we play to 11Nim to 11The first person picks a number between 1-10The second person picks 11 winnerNim to 22The first person picks a number between 1-10Suppose I act secondHow should I act?If I pick 11, then I know I can win, because I can repeat the same set of steps to guarantee victoryExample 3, 11, 19, 22Nim, working backwardsBased on the previous logic, if I can pick numbers that are multiples of 11 less than the winning number, I can guarantee victoryThis is what I will call the “path to victory”Nim, working backwardsExamples of paths to victory99 88 77 66 55 44 33 22 11100 89 78 67 56 45 34 23 12 1In the first game, the first person to act cannot guarantee victory if the other player knows the path to victoryIn the second game, the first player can guarantee victory by choosing 1 and then following the path to victoryPick half the averageRules:Each person picks a number from 0 to 100The person that picks the number closest to half of the average winsIn case of a tie, the winners split the prizePick half the averageIf you assume that each player picks a number randomly between 0 and 100, then I know the average is 50, and I should pick 25However, it would be irrational for anyone to pick a number over 50, since it cannot win Should I pick a number over 25?Pick half the averageI can repeat this thinking an infinite number of times to reach the NEEverybody should pick 0How many people picked…0?A number over
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