Complexity, Appeal and Challenges of Combinatorial Games

1. Complexity, Appeal and Challenges of Combinatorial Games Aviezri S. Fraenkel

2. Outline • Introduction • Play game and ponder • Some definitions, notations, and properties

3. Introduction • We’re not familiar to this field • Games • Playgames • Chess, checker, poker, go, … • Mathgames • Nim, Nimania, Welter’s game, …

4. Nim • Given a finite number of tokens, arranged in piles. A move consists of selecting a pile and removing from it a positive number of tokens, possibly the entire pile • The player making the last move wins, the opponent loses

5. Match 1 • ccli vs shou

6. Nim • Winning strategy: the XOR of the binary representation of the pile sizes is computed. If the XOR is nonzero, the next player can win • Input size: • ni is the size of the ith pile • Strategy computation: linear in input size • Nim is one of the simplest games

7. Nimania • There are some numbers. At the kth stage, a player choose to remove a 1, or break a number m into (k+1) copies of (m-1) • At the first, there is only one number n • The player making the last move wins, the opponent loses

8. Match 2 • kero vs pheno • n = 3

9. Nimania

10. Nimania • If n = 4, the loser can delay the winner so that play lasts over 244 moves • Player I can win for every n 1 • For n  4, the smallest number of move is • For n  4, Player I has a robust winning strategy

11. Definition 4.1 • A game is impartial if the options (moves) of all positions are the same for both players. Otherwise the game is partizan • The game graph of a game  is a digraph G=(V,E), in which every vertex uV represents a game position, and there is a directed edge (u,v)E iff there is a move from u to v in 

12. Welter’s Game • Let’s play on the table, not on the screen  • Try to find the winning strategy • Nim and Nimania are disjoint sum • Welter’s game is not disjoint sum

13. Domineering • A chessboard or other doubly ruled board is tiled with dominoes. Every dominoe covers two adjacent squares • Left tiles vertically, Right horizontally • The player first unable to move loses • Domineering is partizan

14. Grundy’s Game • Given a finite number of piles of finitely many tokens, select a pile and split it into two nonempty piles ofdifferent sizes • The player first unable to move loses • Find the winning strategy if it is not necessary to split into different sizes

15. Poset Game • There are games played on partially ordered set • Example • Chomp

16. Chomp • Two players alternatively move on a given mn matrix of 1’s. For a technical reason there is a 0 at the origin. • A move consists of pointing to some 1, say at location (i, j), and removing the entire north-east sector • The player removing the last 1 wins

17. Match ?

18. Chomp • Player I can win • A neat proof • Could you give me a contructive, polynomial strategy?

19. Notations • A P-position in a game is any position u from which the previous player can force a win • P is the set of all P-position of a game • N-position, tie position, D-position, N, D • F(u) is the set of all (immediate) follows of position u

20. Some Properties • Normal play of a game is when the player making the last move in a game wins; misère play, when the player making the last move loses

21. Superset Game • Put • A move consists of pointing at an as yet unremoved subset and removing it, together with all sets containing it

22. Superset Game

23. Superset Game • For normal play • Conjecture: • If the conjecture is true, then • Why? • Strategy

24. Von Neumann’s Hackendot • It is played on a forest • A player points to an as yet unremoved vertex, and remove the unique path from that vertex to the root of the tree the vertex belongs to

25. Von Neumann’s Hackendot

26. Definition 8 • A subset T of combinatorial games with a polynomial strategy has the following properties. For normal play of every GT, and every position u of G • The P-, N-, D-, or tie-label of u can be computed in polynomial time • The next optimal move can be computed in polynomial time

27. Definition 8 • The winner can consummate a win in at most an exponential number of moves • The subset T is closed under summation, i.e., G1, G2T implies G1+G2T • A game in some such T is called polynomial or efficient or tractable

28. Moore’s Nimk • It’s our last game! • It is a variation of Nim in which up to k piles can be reduced. • Thus Nim is Nim1 • How to modify the XOR strategy?