Chomp on Partially Ordered Sets A Focused Introduction to Combinatorial Game Theory Alexander Clow August 13, 2020 A.Clow August 13, 2020 1 / 12 What is a Combinatorial Game? Partisan vs. Impartial Definition of a combinatorial game1 No Chance Perfect Information Games as Sets of Moves 1Winning ways for your mathematical plays[1] A.Clow August 13, 2020 2 / 12 Examples of Partisan Games A.Clow August 13, 2020 3 / 12 Examples of Impartial Games The Game of Nim th Let there be n 2 N0 piles with xi 2 N stones in the i pile. On their turn a player can remove any amount of stones from a single pile that they wish. The player who takes removes the final stone from the board is the winner. Who wins (x,x) Nim? Is a winner guaranteed? Who wins (3,2,1) Nim? Is a winner guaranteed? A.Clow August 13, 2020 4 / 12 Examples of Impartial Games Poset Chomp Poset Chomp Let (X ; ≤) = P be a partially ordered set. Each turn a player can chose any x 2 X and then remove all x0 2 P such that x ≤ x0. The first player to have no move on their turn loses. Why is Poset Chomp interesting?2 2Computing Winning Strategies for Poset Games[4],A curious nim-type game[2] A.Clow August 13, 2020 5 / 12 Examples of Poset Chomp 3 4 (1) ◦ ◦ (2) ◦ ◦ (3) ◦ (4) ◦ Z D O c O O ◦ ◦ ::: ◦ ◦ ◦ O ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 3Transfinite chomp[3] 4A curious nim-type game[2] A.Clow August 13, 2020 6 / 12 Impartial Combinatorial Games Who wins and how? Zermelo's Theorem5 Theorem If both player play ideally combinatorial games have a set winner and loser or the game will draw. Normal Play P ≡previous player win and N ≡next player win 5Winning ways for your mathematical plays[1] A.Clow August 13, 2020 7 / 12 Sprague Grundy Theorem Sprague Grundy Function G : G 7! N0 Let G be the set of all impartial games Let A 2 G,G(A) = MinfG(A0): 8 A0 one move from AgC Let ; ≡the empty-game, G(;) = 0 Sprague Grundy Theorem6 Theorem Every impartial game A is equivalent in terms of the disjoint sum of games to a pile in Nim of height m, if and only if G(A) = m. =) 8A; A ≡ P () G(A) = 0 6Winning ways for your mathematical plays[1] A.Clow August 13, 2020 8 / 12 Sum of Disjoint Impartial Games Nim-Sums≡ L Adding Impartial Games7 Let A; B be impartial games. A + B ≡ A [· B G(A + B) = G(A) ⊕ G(B) Recall games are sets of moves. Nim-Sums≡ L ⊕ ≡ to Binary XOR 7Winning ways for your mathematical plays[1] A.Clow August 13, 2020 9 / 12 Poset Chomp Novel Result Let S be the set of all sets of non-negative integers Φx : S 7! S, x 2 N0 C Φx (S) = Φx−1(S [ fMin(S) g) and Φ0(S) = S Theorem Let P; A; B be partially ordered sets such that P = A [ B where for all a 2 A and b 2 B b < a, 0 0 C G(P) = MinfΦG(A)fG(B ): B − b≤gg A.Clow August 13, 2020 10 / 12 Poset Chomp Intuition + Examples ◦ ◦ ◦ (1) ; c (2) ; c (3) O ◦ ◦ ◦ ◦ ◦ ◦ c ; O c O ◦ ◦ ◦ ◦ ◦ ◦ ◦ ; c c O ; c O ; ◦ c ; ◦ ◦ ci ;5 ◦ ◦ ◦ ◦ ; c O O ◦ c ; ◦ ◦ ◦ ; ◦ c ◦ ◦ A.Clow August 13, 2020 11 / 12 Bibliography John H Conway, Richard K Guy, and Elwyn R Berlekamp. Winning ways for your mathematical plays. 2003. David Gale. A curious nim-type game. The American Mathematical Monthly, 81(8):876{879, 1974. Scott Huddleston and Jerry Shurman. Transfinite chomp, volume 42. MSRI Publications, 2002. Craig Wilson. Computing Winning Strategies for Poset Games. PhD thesis, 2008. A.Clow August 13, 2020 12 / 12.