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Complexity, Appeal and Challenges of Combinatorial Games

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Complexity, Appeal and Challenges of Combinatorial Games

Aviezri S. Fraenkel

- Introduction
- Play game and ponder
- Some definitions, notations, and properties

- We’re not familiar to this field
- Games
- Playgames
- Chess, checker, poker, go, …

- Mathgames
- Nim, Nimania, Welter’s game, …

- Playgames

- 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

- ccli vs shou

- 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

- 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

- kero vs pheno
- n = 3

- 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

- 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 uV represents a game position, and there is a directed edge (u,v)E iff there is a move from u to v in

- 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

- 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

- 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

- There are games played on partially ordered set
- Example
- Chomp

- Two players alternatively move on a given mn 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

- Player I can win
- A neat proof
- Could you give me a contructive, polynomial strategy?

- 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

- 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

- Put
- A move consists of pointing at an as yet unremoved subset and removing it, together with all sets containing it

- For normal play
- Conjecture:
- If the conjecture is true, then
- Why?
- Strategy

- 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

- A subset T of combinatorial games with a polynomial strategy has the following properties. For normal play of every GT, 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

- The winner can consummate a win in at most an exponential number of moves
- The subset T is closed under summation, i.e., G1, G2T implies G1+G2T

- 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?