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Topic: Combinatorial game theory

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	Combinatorial Game Theory Background
		The key to successful mathematical analysis of a combinatorial game as turned out to be the question of whether or not the positions in the game tend to decompose into sums of simpler positions.
		Games on one side of the chasms are fruitful targets for paper-and-pencil mathematical analysis, based on an ever-growing base of general theorems.
		As I use the term, “combinatorial game theory” differs from artificial intelligence in the same primary respect that mathematics differs from engineering: the emphasis is on conclusions which are provably correct rather than on conclusions which are plausible or good enough to be acceptable in some imprecise sense.
	math.berkeley.edu /~berlek/cgt/cgt-info.html (2607 words)

	Combinatorial game theory - Wikipedia, the free encyclopedia
		Combinatorial game theory (CGT) is a mathematical theory that only studies two-player games which have a position which the players take turns changing in defined ways or moves to achieve a defined winning condition.
		CGT does not study games of chance (like poker), but restricts itself to games whose position is public to both players, and in which the set of available moves is also public.
		CGT should not be confused with another mathematical theory, traditionally called game theory, used in the theory of economic competition and cooperation.
	en.wikipedia.org /wiki/Combinatorial_game_theory (1488 words)

	Game theory - Wikipedia, the free encyclopedia
		Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns.
		Game theorists may assume players always act rationally to maximize their wins (the Homo economicus model), but real humans often act either irrationally, or act rationally to maximize the wins of some larger group of people (altruism).
		Game theory experienced a flurry of activity in the 1950s, during which time the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value were developed.
	en.wikipedia.org /wiki/Game_theory (3878 words)

	Combinatorial game theory - Simple English Wikipedia
		Combinatorial game theory is a mathematical theory of games.
		It might seem as if few games like that would be interesting--but it turns out that a large number of new and old games can be analyzed with CGT.
		For example, in the game of chess, there is a usual starting setup.
	simple.wikipedia.org /wiki/Combinatorial_game_theory (431 words)

	Combinatorial Game Theory
		Combinatorial Game Theory studies strategies and mathematics of two-player games of perfect knowledge such as chess or go (but often either concentrating instead on simpler games such as nim, or solving endgames and other special cases).
		An important distinction between this subject and classical game theory (a branch of economics) is that game players are assumed to move in sequence rather than simultanously, so there is no point in randomization or other information-hiding strategies.
		In this game, a position is represented by a number, and one moves by either adding or subtracting the largest prime number less than or equal to the position.
	www.ics.uci.edu /~eppstein/cgt (1515 words)

	urticator.net - Combinatorial Game Theory
		Combinatorial game theory, not to be confused with regular game theory, is the kind of mathematics used to describe games in which the goal is to make the last move.
		Strictly speaking, then, it applies only to games like Nim and Sprouts (see Macroscope for the latter), but it turns out that the same theory can be adapted to other games as well, such as Dots and Boxes and even the endgame in go.
		There's even a natural operation on games that acts as addition, namely, playing both games at the same time, with a move in the composite game being a move in either one of the component games.
	www.urticator.net /essay/0/11.html (497 words)

	Sensei's Library: Combinatorial Game Theory
		CGT was in part inspired by Go, since many Go end positions (yose) are combinations of independent regions of play.
		Each such position is a combinatorial game, which can be added to or subtracted from other such games.
		Numbers are special cases of combinatorial games: Go players can take that as meaning that some positions are already secure territory.
	senseis.xmp.net /?CombinatorialGameTheory (412 words)

	winways
		Most games are finite, but imagine chess as played without stalemate rules or a version of tic-tac-toe where after a draw the board is cleared and the game begins again; these would be infinite games.
		Dynamic games involve the taking of turns, as opposed to static games, where players choose independently and their choices are revealed simultaneously.
		Much of game theory is about seeking ways to lower this number or to circumvent the need for a tree at all.
	www.mathpuzzle.com /winways.htm (1201 words)

	Combinatorial Games (Erik Demaine)
		Two versions of ``Playing Games with Algorithms: Algorithmic Combinatorial Game Theory'' are available: the full draft (recommended), and a shorter version appearing in the 26th Symposium on Mathematical Foundations in Computer Science.
		Triangulation games: A variety of geometric games involving the construction, transformation, or marking of the edges of a planar triangulation.
		Tetris: The classic computer game in which tetrominoes (pieces made up of 4 unit squares) fall one at a time into a rectangular board, the player can slide the piece left or right during the fall, and any completely filled rows are erased.
	theory.lcs.mit.edu /~edemaine/games (840 words)

	CGT Becomes Hard Currency
		CGT has something new to say about open-ended plays, and has revealed fine structure showing how delicate games can be if they depend on the last point played.
		A start on describing what is seen in real games is the combination of disjunction (two or more games played side-by-side in a modular way), and a rigorous idea of an ambient temperature, relative to which the hot spots and low-priority areas in the overall game may be charted.
		In recent trials of this game with top pros Jiang Zhujiu and his wife Rui Naiwei (on the basis of her current games in South Korea the highest-ever achiever among women in mind sports), he has started to collect information relating the actual judgements of very strong players over the board with the basic theory.
	gobase.org /studying/articles/matthews/CGT (1259 words)

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		In essence, Combinatorial Game Theory is study of two-person games.
		Combinatorial games do not allow dice, the shuffling of cards, or any other devices which lead to the need for probabilities and distrubutions.
		All combinatorial games require for all game data to be accessible to both players.
	cgt.calculusfairy.com /CGTIntro.html (511 words)

	urticator.net - Game Theory
		Although game theory is a theory of games, it's sufficiently generalized that the games might not be immediately recognizable as such—they might be, for example, economic decisions.
		From what I've read of Theory of Games and Economic Behavior, which I believe is the original work on game theory, it looks like there's a definite method of solution.
		Anyway, the example I finally settled on is the game Undercut, which is both defined and analyzed in the eponymous chapter of Metamagical Themas.
	www.urticator.net /essay/2/218.html (721 words)

	[No title]
		Before understanding Combinatorial Game Theory, one must first define what Game Theory is. A branch of mathematics and logic which deals with the analysis of games (i.e., situations involving parties with conflicting interests).
		Game Theory allows the player or parties to have a better understanding of what the rules and boundaries of a given situation of games are and developing strategies for players.
		Combinatorial Game Theory is defined as the studies and strategies of two or more player games with the players having complete knowledge of the game.
	www.saintjoe.edu /~karend/m441/RomanFinalPaper.doc (3411 words)

	IBFI Schloss Dagstuhl - Dagstuhl Seminar 02081
		The combinatorial game theory community has studied games extensively, resulting in powerful tools for their analysis, like the notion of game-theoretic value.
		Game theory could benefit from applying algorithmic techniques to games with known outcomes but no known efficient strategies, e.g., Hex and poset games such as Chomp.
		The goal of this seminar is to bring these two communities together, to advance the area of algorithmic combinatorial game theory from infancy to maturity.
	www.dagstuhl.de /02081 (432 words)

	The Math Forum - Math Library - Game Theory
		Game theory is an interdisciplinary approach to studying human behavior that touches on computer science, social sciences, and behavioral sciences.
		Combinatorial Game Theory studies strategies and mathematics of two-player games of perfect knowledge such as chess or go (but often either concentrating instead on simpler games such as nim, or solving endgames...more>>
		A game that has been and continues to be studied by people in a variety of disciplines, ranging from biology through sociology and public policy.
	mathforum.org /library/topics/game_theory (2290 words)

	Mathematical Games, Toys, and Puzzles
		After taking Elwyn Berlekamp's combinatorial games class, I wrote a couple of papers, one on Toads and Frogs and another on Sowing Games.
		Turnablock, one of many games invented and analyzed by John Conway, implemented as a Java applet by Ken Shirriff.
		The Games Domain is mostly about PC video games (yawn), but there are a few pointers to other "abstract" games pages.
	compgeom.cs.uiuc.edu /~jeffe/mathgames.html (1713 words)

	Al Roth's game theory, experimental economics, and market design page (Site not responding. Last check: 2007-10-15)
		Game theory is the part of economics concerned with the detailed rules and procedures of economic institutions.
		The theory of the complaint is that a match holds down wages for residents and fellows.
		Combinatorial Game Theory: Some game theory with connections to operations research and computer science can be found at Combinatorial Game Theory maintained by David Eppstein at UC Irvine.
	kuznets.fas.harvard.edu /~aroth/alroth.html (6628 words)

	Combinatorial Game Theory (Site not responding. Last check: 2007-10-15)
		Combinatorial game theory is a mathematical theory that can solve difficult Go endgame problems.
		This theory has a branch called thermography, which is useful for playing endgames well with only a moderate amount of analysis.
		He has written a C library for combinatorial games and late Go endgames, which I used as subroutines in my program to solve Go endgames.
	www.cs.ualberta.ca /~mmueller/cgt (185 words)

	combinatorial games (Site not responding. Last check: 2007-10-15)
		Last July, at the Combinatorial Games Workshop in Berkeley, David Blackwell suggested that most of mathematics may be chaotic, and that it is only the small part where we recognize patterns that we actually call mathematics.
		Combinatorial game theory tends to foster such a view.
		Combinatorial game theory is just one example of what unfettered mathematical imagination is capable of creating...
	www.plambeck.org /oldhtml/mathematics/games (396 words)

	ipedia.com: Combinatorial game theory Article (Site not responding. Last check: 2007-10-15)
		Combinatorial game theory is a mathematical theory of games, which while part of game theory in a broad sense has its own tradition going back to the solution of Nim.
		The elements of are called games and the convention here is that they would be denoted by the upper case Latin letters G,H,K,...
		The set of nimbers is defined as the smallest subcollection containing 0 and containing for every G in the subcollection.
	www.ipedia.com /combinatorial_game_theory.html (457 words)

	Amazon.com: So You'd Like to... Learn about Combinatorial Game Theory (Site not responding. Last check: 2007-10-15)
		Combinatorial Game Theory is used to analyze the outcomes of "perfect-information" games like Nim and Tic-tac-toe.
		The theory stems from John Conway and his creation of surreal numbers, presented in the book, 'On Numbers and Games'.
		Scientific papers on combinatorial games have become more prominent, and some of the "must-read" ones are collected in the journals: 'Games of No Chance (Mathematical Sciences Research Institute Publications)' and the appropriately-named 'More Games of No Chance (Mathematical Sciences Research Institute Publications)'.
	www.amazon.com /exec/obidos/tg/guides/guide-display/-/2SON8U8GSZE56 (487 words)

	Combinatorial Game Suite (Site not responding. Last check: 2007-10-15)
		Combinatorial Game Suite is an open-source program to aid research in combinatorial game theory.
		Combinatorial Game Suite is completely open-source, and source code is available; see the download page for details.
		Combinatorial Game Suite was designed and developed by Aaron Siegel (http://math.berkeley.edu/~asiegel/).
	cgsuite.sourceforge.net (264 words)

	Articles - Game theory (Site not responding. Last check: 2007-10-15)
		First developed as a tool for understanding economic behavior, game theory is now used in many diverse academic fields, ranging from biology to philosophy.
		Although Cournot´s analysis is more general than Waldegrave´s, game theory did not really exist as a unique field until John von Neumann (the "von" was from an enoblement by the Hungarian government) published a series of papers in 1928.
		Von Neumann was a brilliant mathematician whose work was far-reaching from set theory to his calculations that were key to development of both the Atom and Hydrogren bombs and finally to his work developing computers.
	www.totalorange.com /articles/Game_theory (1498 words)

	Aviezri Fraenkel's Home Page
		A.S. Fraenkel and A. Kontorovich, The Sierpinski sieve of Nim-varieties and binomial coefficients.
		A.S. Fraenkel, Combinatorial game theory foundations applied to digraph kernels, Electronic J. of Combinatorics 4(2) (1997) R10, 17pp.
		A dynamic bibliography on Combinatorial Games, currently containing 1160 items is a Dynamic Survey (D2, originally 1994) in the Dynamic Surveys section of the Electronic Journal of Combinatorics.
	www.wisdom.weizmann.ac.il /~fraenkel (1754 words)

	Citebase - Playing Games with Algorithms: Algorithmic Combinatorial Game Theory
		Authors: Demaine, Erik D. Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open.
		In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the gameboard is cleared and all pieces above it drop by one row.
		The early game can be treated as a case of Random Sequential Adsorption, with a jamming transition that marks the beginning of the end-game.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:cs/0106019 (1577 words)

	Combinatorial games procedures for Maple (Site not responding. Last check: 2007-10-15)
		The current predominant combinatorial games toolkit is Aaron Siegel's cgsuite.
		Games that are numbers are displayed as Maple displays numbers of its internal numeric type.
		Infinitesimal games that are sums of ups and stars are recognized and displayed as such.
	members.shaw.ca /a00/combfunctions.html (602 words)

	CCS Math 10: Combinatorial Game Theory
		All games bright and beautiful” by John Conway is a sort of summary of
		Misère games and misère quotients” by Aaron Siegel is a good place to start if you are interested in misère games.
		Combinatorial Game Suite is a Java program written by Aaron Siegel that helps you to find values of certain games.
	www.albanyconsort.com /simon/cgt.html (440 words)

	GameSites : Games - Paper and Pencil - Dots and Boxes
		Category: Games » Paper and Pencil » Dots and Boxes
		Reviews standard rules, and introduces the analogous game on a hexagonal board.
		Streaming-video 30-minute talk held at MSRI during the Combinatorial Game Theory Research Workshop, July 24-28, 2000.
	gamesites.bluechillies.com /section/416167 (102 words)

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