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 ∈ N0 piles with xi ∈ 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 ∈ X and then remove all x0 ∈ 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 ∈ G,G(A) = Min{G(A0): ∀ A0 one move from A}C 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.

=⇒ ∀A, 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 ∈ N0

C Φx (S) = Φx−1(S ∪ {Min(S) }) and Φ0(S) = S

Theorem Let P, A, B be partially ordered sets such that P = A ∪ B where for all a ∈ A and b ∈ B b < a,

0 0 C G(P) = Min{ΦG(A){G(B ): B − b≤}}

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