Combinatorial Game Theory: Impartial and Partisan Games, Slides of Game Theory

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Combinatorial Game Theory

By Adithya M S

What is Combinatorial Game Theory?

• A branch of mathematics that studies strategic games with finite, discrete moves
• Focuses on games with perfect information and no randomness in involved In combinatorial games:
• There are two players
• Players take turns to make moves
• NO chance element is involved, and we know the winner as long as the game ends As per our convention, we shall take the players to be Left and Right Examples of combinatorial games include Chess, Nim and Chomp Non-examples include Snakes-and-Ladders, Ludo, Rummy etc

Types of Combinatorial Games

We have two types of combinatorial games:

• Impartial
• Partisan In impartial games, the set of available moves for both the players is the same. For example: Nim In partisan games, each player has a different set of moves from a given position. Chess is an example of a partisan game as a white player can only move white pieces and the black player only black in a given turn https://www.cs.cmu.edu/afs/cs/academic/class/15859- s05/www/ferguson/comb.pdf

Outcomes of a partisan game

Lessons in Play: An Introduction to Combinatorial Games

Game Trees

• A game tree is a graphical representation of all possible positions and moves in a combinatorial game
• Each node in the tree represents a game position, and each edge represents a possible move
• The game tree shows all possible paths the game can take, from the initial position to the final position Lessons in Play: An Introduction to Combinatorial Games

Calculating the outcome of a position

The outcome class of a game may be decided recursively by the outcome classes of its options One can refer to the following table: It is not difficult to see that in an impartial game, we have ONLY P-positions and N-positions. No position in an impartial game can have outcome L or R. Lessons in Play: An Introduction to Combinatorial Games

Poset Games with examples

• A poset game is a combinatorial game played on a partially ordered set (poset)
• Players take turns removing an element and all elements greater than or equal to it Poset in Nim
• In Nim, the poset is a set of piles of stones partially ordered by inclusion
• Removing stones from a pile corresponds to removing elements in the poset Poset in Chomp
• In Chomp, the poset is a grid of squares, partially ordered by the “above” and “to the left” relations
• Removing a square corresponds to removing all squares above and to the right of it from the poset https://en.wikipedia.org/wiki/Poset_game

General result on Poset Games

If P is a poset with a greatest element g, then the game on poset P is always won by the first player

• First player removes greatest element. If winning move, done
• If not, second player responds with move e
• First player can "steal" second player's move e
• By doing so, first player puts second player in same position as original first player
• Second player cannot win from this position https://aeb.win.tue.nl/games/chomp.html

Graph Games and Sprague Grundy Theorem

Graph Games

• A graph game is a combinatorial game played on a directed graph.
• Players take turns moving a token along the edges of the graph.
• The game ends when a player cannot make a move Sprague-Grundy Theorem
• Assigns a Grundy value to each position in the game
• The Grundy value of a position is the smallest non-negative integer that is not the Grundy value of any of its options (i.e., positions that can be reached from the current position)
• The theorem states that a player can win from a position if and only if its Grundy value is non-zero
• Allows for the solution of many combinatorial games, including Nim and other impartial games
• The Grundy value of a disjunctive sum of games is the nim sum of their Grundy values

Algebra of Games

A game (position) G is defined by its options G = {GL | GR} where GL and GR are the set of Left and Right options respectively. A Left (Right) option of a game position refers to a position to which Left (Right) may move to starting from that position Often, positions decompose into components during game play, in which case players need to choose a component to play in and a disjunctive sum is defined accordingly In other words, the disjunctive sum of two games G and H is the game where a player chooses one of the games and makes a move in it (expressed as G + H) The negative of a game G (written as - G) (also defined recursively later) is the game G with the roles of the players interchanged (The moves that can be made by Left may be played by Right and vice versa)

Algebra of Games (3)

Consider a set S containing positions from a ruleset. This set is referred to as a hereditarily closed set of positions of a ruleset (HCR) if it remains closed under Left or Right options—meaning, if s ∈ S then every option of s is also in S.

Values of Games

This equality relation between games induce equivalence classes and there is a "smallest" game G 0 that is a representative of this equivalence class of a game (say G). This smallest game is called the canonical form or game value of G There are two important rules to simplify a game G to its canonical form called Domination and Simplicity rules

CHOMP as a divisor game

• Consider an (n+1) x (m+1) chocolate bar with blocks labeled (i, j).
• Each block corresponds to a number p1^i * p2^j, where p1 and p2 are prime divisors.
• A block B1 is "less than or equal to" B2 if the number labeling B1 divides the number labeling B
• Chomp can be viewed as a game of removing divisors, where a player's move corresponds to removing a block and all its multiples

BLUE-RED Hackenbush

BLUE-RED Hackenbush is a game where players start with drawing a ground line and several line segments, that are directly or via a chain of other line segments, connected to the ground, Players take turns to cut (erase) a line segment of their after which line segments no longer connected to the ground fall apart automatically. The first player unable to move loses One of the players can ONLY cut the red segments whereas the other only blue