155_Fall_18 (2).pdf - Game Theory Homework#2 All the problems are worth 4 points each and will be graded on a 0\/1\/2\/3\/4 scale Due on Wednesday before

# 155_Fall_18 (2).pdf - Game Theory Homework#2 All the...

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Game Theory Homework #2, 09/11/2018 All the problems are worth 4 points each and will be graded on a 0/1/2/3/4 scale. Due on Wednesday 09/19/2018 before 11:59 pm to be uploaded via Gradescope. 1. Let G be a progressively bounded impartial combinatorial game under normal play. Its Sprague-Grundy functiongisdefined recursively as follows:g(x) = mex{g(y) :xyis a legal move.}wheremexof a set of numbers, is the minimum excluded value function (smallest non-negative number not in the set).Some examples are as follows:mex{0,1,3,4}= 2,mex{2,5,7}= 0.Note that the Sprague-Grundy value of any terminal position ismex() = 0.Prove thatxPiffg(x) = 0.Now consider the sumGof two progressively bounded impartial combinatorial game under normal play,G1andG2.Letg, g1andg2be the respective Sprague-Grundy functions ofG, G1, G2respectively.Show thatg(x1, x2) =g1(x1)g2(x2).2. Find the Sprague-Grundy function for the Nim game(n1, n2, . . . nk).Consider a game of Nim with four piles, of sizes9,10,11,12.Is this position a win for the next player or the previous player (assuming optimal play)? Describe all the winning