Great Theoretical Ideas In Computer Science V Adamchik D Sleator Lecture 10 CS 15 251 Feb 11 2010 Spring 2010 Plan References Introduction to Impartial Combinatorial Games Game Theory by T Ferguson Carnegie Mellon University Mathematical Games Download from http www math ucla edu tom Game Theory Contents html Related courses 15 859 21 801 Mathematical Games Look for it in Spring 11 1 A Take Away Game Two Players 1 and 2 A move consists of removing one two or three chips from the pile Players alternate moves with Player 1 starting 21 chips Player that removes the last chip wins Which player would you rather be What if the last player to move loses If there is 1 chip the player who moves next loses If there are 2 3 or 4 chips left the player who moves next can win by leaving only 1 In this case 1 5 9 13 are a win for the second player Try Small Examples If there are 1 2 or 3 only player who moves next wins 0 4 8 12 16 are target positions if a player moves to that position they can win the game If there are 4 chips left player who moves next must leave 1 2 or 3 chips and his opponent will win With 5 6 or 7 chips left the player who moves next can win by leaving 4 chips Combinatorial Games A set of positions Two players Rules specify for each player and for each position which moves to other positions are legal moves The players alternate moving A terminal position in one in which there are no moves The game ends when a player has no moves The game must end in a finite number of moves No draws 21 chips Therefore with 21 chips Player 1 can win Normal Versus Mis re Normal Play Rule The last player to move wins Mis re Play Rule The last player to move loses A Terminal Position is one where neither player can move anymore 2 What is Omitted Impartial Versus Partizan P Positions and N Positions No random moves This rules out games like poker No hidden state This rules out games like battleship No draws in a finite number of moves This rules out tic tac toe A combinatorial game is impartial if the same set of moves is available to both players in any position A combinatorial game is partizan if the move sets may differ for the two players In this lecture we ll study impartial games Partizan games will not be discussed What s a P Position 0 4 8 12 16 are Ppositions if a player moves to that position they can win the game Positions that are winning for the Previous player the player who just moved That means For any move that N makes There exists a move for P such that For any move that N makes There exists a move for P such that 21 chips is an N position 21 chips There exists a move for P such that P Position Positions that are winning for the Previous player the player who just moved Sometimes called LOSING positions N Position Positions that are winning for the Next player the player who is about to move Sometimes called WINNING positions P positions and N positions can be defined recursively by the following 1 All terminal positions are Ppositions normal winning condition 2 From every N position there is at least one move to a P position 3 From every P position every move is to an N position There are no possible moves for N 3 Chomp Show That This is a P position Two player game where each move consists of taking a square and removing it and all squares to the right and above Player who takes position 1 1 loses Show That This is an N position P position N Positions Let s Play I m player 1 Mirroring is an extremely important strategy in combinatorial games No matter what you do I can mirror it 4 Theorem Player 1 can win in any square starting position of Chomp Theorem Every rectangle is a N position The Game of Nim Proof Consider this position Two players take turns moving Proof The winning strategy for player 1 is to chomp on 2 2 leaving only an L shaped position Then for any move that Player 2 takes Player 1 can simply mirror it on the flip side of the L Analyzing Simple Positions This is either a P or an N position If it s a P position then the original rectangle was N If it s an Nposition then there exists a move from it to a Pposition X But by the geometry of the situation X is also available as a move from the starting rectangle It follows that the original rectangle is an N position One Pile Nim x y z Winner is the last player to remove chips A move consists of selecting a pile and removing chips from it you can take as many as you want but you have to at least take one In one move you cannot remove chips from more than one pile Two Pile Nim P positions are those for which the two piles have an equal number of chips x y z We use x y z to denote this position 0 0 0 is a P position What happens in positions of the form x 0 0 The first player can just take the entire pile so x 0 0 is an N position If it is the opponent s turn to move from such a position he must change to a position in which the two piles have different number of chips From a position with an unequal number of chips you can easily go to one with an equal number of chips perhaps the terminal position 5 Nim Sum The nim sum of two non negative integers is their addition without carry in base 2 For any non negative integer x Cancellation Property Holds If x y x z We will use to denote the nim sum 2 3 10 2 11 2 01 2 1 x x 5 3 101 2 011 2 110 2 6 Then x x y x x z 0 So y z 7 4 111 2 100 2 011 2 3 is associative a b c a b c is commutative a b b a Bouton s Theorem A position x y z in Nim is a P position if and only if x y z 0 Proof Let Z denote the set of Nim positions with nim sum zero Let NZ denote the set of Nim positions with non zero nim sum We prove the theorem by proving that Z and NZ satisfy the three conditions of P positions and N positions 1 All terminal positions are in Z The only terminal position is 0 0 0 2 From each position in NZ there is a move to a position in Z 001010001 100010111 111010000 3 Every move from a …
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