By Ferguson T.S.
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Additional info for Game theory
3 Pure Strategies and Mixed Strategies. 4 The Minimax Theorem. 5 Exercises. 2. Matrix Games. Domination. 1 Saddle Points. 2 Solution of All 2 by 2 Matrix Games. 3 Removing Dominated Strategies. 4 Solving 2 × n and m × 2 Games. 5 Latin Square Games. 6 Exercises. 3. The Principle of Indiﬀerence. 1 The Equilibrium Theorem. 2 Nonsingular Game Matrices. 3 Diagonal Games. 4 Triangular Games. 5 Symmetric Games. 6 Invariance. 7 Exercises. 4. Solving Finite Games. 1 Best Responses. 2 Upper and Lower Values of a Game.
If (5) represents a position in Turning Corners, we can ﬁnd the SG-value easily as 3 ⊕ 1 ⊕ 6 = 4. This is therefore an N-position, that can be moved to a P-position by moving the SG-value 6 to an SG-value 2. There is a unique way that this may be done, namely by turning over the four coins at (3,3), (3,4), (4,3) and (4,4). 3 Nim Multiplication. 2 may seem rather haphazard, but such an appearance is deceptive. In fact, the function g deﬁned by (6) may be considered as a form of nim multiplication.
4. Solve the Kayles problem of Dudeney and Loyd. 1 The Kayles problem of Dudeney and Loyd (a) Show this is an N-position. 1. (b) Find a winning move. Which pin(s) should be knocked down? html and beat the computer. 5. 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.
Game theory by Ferguson T.S.