Game theory - download pdf or read online
By Ferguson T.S.
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Extra resources for Game theory
That is, at least one edge must be removed. Last player to move wins. Investigate this game. For example, (a) Find the Sprague-Grundy value of Sn , the star with n points. ) (b) Find the Sprague-Grundy value of Ln , the line of n edges and n + 1 vertices (at least for small values of n). (c) Find the Sprague-Grundy value of Cn, the circular graph of n vertices and n edges. (d) Find the Sprague-Grundy value of a double star. ) I – 27 (e) Who wins the 3 by 3 square lattice, (i)? Who wins the tic-tac-toe board, (ii)?
Suppose at each turn a player may (1) remove one or two chips, or (2) remove one chip and split the remaining chips into two piles. (a) Find the Sprague-Grundy function. (b) Suppose the starting position consists of one pile with 15 chips. Find an optimal ﬁrst move. I – 26 6. Suppose that at each turn a player may (1) remove one chip if it is a whole pile, or (2) remove two or more chips and, if desired, split the remaining chips into two piles. Find the Sprague-Grundy function. 7. Suppose that at each turn a player may select one pile and remove c chips if c = 1 (mod 3) and, if desired, split the remaining chips into two piles.
10. Grundy’s Game. 4, for a pile of n chips for n = 1, 2, . . , 13. (b) In Grundy’s game with three piles of sizes 5, 8, and 13, ﬁnd all winning ﬁrst moves, if any. 11. A game is played on a ﬁnite (undirected) graph as follows. Players alternate moves. A move consists of removing a vertex and all edges incident to that vertex, with the exception that a vertex without any incident edges may not be removed. That is, at least one edge must be removed. Last player to move wins. Investigate this game.
Game theory by Ferguson T.S.