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Topic: Minimax algorithm

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	Algorithms
		Search algorithms tend to utilize a cause-and-effect concept--the search considers each possible action available to it at a given moment; it then considers its subsequent moves from each of those states, and so on, in an attempt to find terminal states which satisfy the goal conditions it was given.
		For two player games, the minimax algorithm is such a tactic, which uses the fact that the two players are working towards opposite goals to make predictions about which future states will be reached as the game progresses, and then proceeds accordingly to optimize its chance of victory.
		The theory behind minimax is that the algorithm's opponent will be trying to minimize whatever value the algorithm is trying to maximize (hence, "minimax").
	www.stanford.edu /~msirota/soco/minimax.html (874 words)

	Minimax algorithm
		The minimax algorithm is a recursive algorithm for choosing the next move in a two-player game.
		The effective branching factor[?] of the tree is the average number of children of each node (i.e., the average number of legal moves in a position).
		The performance of the naive minimax algorithm may be improved dramatically, without affecting the result, by the use of alpha-beta pruning.
	www.ebroadcast.com.au /lookup/encyclopedia/mi/Minimax.html (372 words)

	Minimax - Wikipedia, the free encyclopedia
		Minimax (sometimes minmax) is a method in decision theory for minimizing the maximum possible loss.
		The performance of the naïve minimax algorithm may be improved dramatically, without affecting the result, by the use of alpha-beta pruning.
		Minimax theory has been extended to decisions where there is no other player, but where the consequences of decisions depend on unknown facts.
	en.wikipedia.org /wiki/Minimax (1213 words)

	Intro to CogSci Minimax Exercise
		Minimax is an algorithm for playing two-player games (it generalizes to multi-player games as well).
		In minimax, a heuristic is a number that represents the relative strategic value of a game state from the perspective of the computer.
		Minimax suffers from exponential explosion of number of nodes or states - that is, the number of states to consider goes up exponentially with the number of ply.
	www.cs.vassar.edu /faculty/welty/oldcourses/cogsci100.98f/minimax.html (946 words)

	Minimax and Alpha-Beta Pruning
		Notice that the maximin and minimax values are symmetric; if I define the payoff relative to me instead of you then I should choose the maximin solution and you should choose the minimax solution.
		Minimax is a useful decision concept and it is straightforward to find the minimax solution and minimax values when a game is in a simple matrix form like the example above; simply do an exhaustive search through all possibilities.
		The basic algorithm is for me to maximize over my first move over the minimum of your first move over the maximum over my second move over the minimum over your second move, and so on.
	students.cs.byu.edu /~cs670ta/Lectures/Minimax.html (1908 words)

	Alpha-beta pruning - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-06)
		Alpha-beta pruning is a search algorithm that reduces the number of nodes that need to be evaluated in the search tree by the minimax algorithm.
		The algorithm maintains two values, alpha and beta, which represent the minimum score that the maximizing player is assured of and the maximum score that the minimizing player is assured of respectively.
		Since the minimax algorithm and its variants are inherently depth-first, a strategy such as iterative deepening is usually used in conjunction with alpha-beta so that a reasonably good move can be returned even if the algorithm is interrupted before it has finished execution.
	en.wikipedia.org /wiki/Alpha-beta_pruning (992 words)

	Minimax and Alpha-Beta Template
		The minimax algorithm is a method of selecting the best choice of action in a situation, or game, where two opposing forces, or players, are working toward mutually exclusive goals, acting on the same set of perfect information about the outcome of the situation.
		The minimax algorithm is a recursive algorithm that takes as arguments the current layout of the game board and a parity number which reflects whether you want a minimax or maximin.
		Since the amount of work a minimax search generates increases exponentially as a move is examined to a greater depth, to reduce the time required for the search it must be restricted so that no time is wasted searching moves that are obviously bad for the player.
	www.cs.caltech.edu /~petrovic/games/archex/othellodir/node2.html (952 words)

	Linux Only - Ataxx Algorithms
		Minimax tries to maximize the advantage for the current player, while minimizing the advantage for the other player.
		In minimax, all heuristic values of the nodes on a certain ply depth are evaluated.
		A disadvantage of the minimax algorithm is that each board state has to be visited twice: one time to find its children and a second time to evaluate the heuristic value.
	www.linuxonly.nl /docs/ataxx/?p=3 (452 words)

	Scotland Yard AI: Minimax (Site not responding. Last check: 2007-11-06)
		The point of the minimax algorithm is to pick a players best move, assuming that it will be followed by an opponent move.
		The minimax algorithm calculates all of the possible moves that can be made by a player based on some game specific rules, then given all possible moves it calculates all possible moves of the opponent as a result of each of the first players moves.
		It is the job of minimax algorithm to pass all legal states to the heuristic function.
	www.ccs.neu.edu /home/dskippy/code/scotland-yard/minimax.html (408 words)

	The Minimax Algorithm
		The Minimax algorithm is a method of searching the game-state graph in order to choose the move that will result in the best possible outcome from a given instance.
		Although the Minimax algorithm is a top-down algorithm, it is easier to understand from the bottom up.
		One problem with the standard Minimax algorithm is the amount of time it takes to make a decision.
	zoo.cs.yale.edu /classes/cs490/00-01b/kerner.matthew.mmk29/quarto/node7.html (367 words)

	generation5 - Applying Artificial Intelligence, Search Algorithms and Neural Networks to Games
		Minimax can allow a game to look ahead at the possible moves that a human player can make and therefore it can come up with a move that allows the computer to win.
		Minimax can also be used effectively if the depth of the search is limited and the game has a limited number of solutions, such as chess.
		Where the minimax tries to anticipate the moves of a player, it does not ‘learn’ how to play the game and doesn’t account for a player who may not want to minimise the score of their opponent.
	www.generation5.org /content/2003/KellyMiniPaper.asp?Print=1 (3100 words)

	Computer Chess
		The minimax algorithm is primarily used for seraching moves on the board.
		Other algorithms such as the alpha-beta algorithm are powerful additions to the minimax algorithm.The minimax algorithm serves as the foundation of chess programs.
		According to the minimax algorithm, MAX's aim is to choose a move with highest value in the move tree;on the other hand MIN's goal is to do exactly the opposite.
	www.ccs.neu.edu /groups/honors-program/freshsem/19951996/rsuchak (1259 words)

	Aske Plaat: MTD(f), a new chess algorithm
		The better than first guess is, the more efficient the algorithm will be, on average, since the better it is, the less passes the repeat-until loop will have to do to converge on the minimax value.
		George Stockman's best-first minimax algorithm that promised to be more efficient than AlphaBeta.
		Best-First Fixed-Depth Minimax Algorithms", which also deals with the relation of other MTD instances to SSS-like best-first minimax* algorithms.
	www.cs.vu.nl /~aske/mtdf.html (2779 words)

	Off-the-shelf AI : Plug-in Minimax - AI Factory newsletter article Autumn 2005
		The basis of the Minimax algorithm is simply to minimise the maximum possible loss by choosing a move which offers your opponent the minimum gain in replying.
		However this is misleading as the behaviour of this simple algorithm shows great complexity, perhaps analogous to the way that fractals generate complex structures with just a very simple algorithm.
		The minimax used by AI Factory is highly developed, from 25 years direct practical experience, backed up by the support of thousands of academic papers published on this topic.
	freespace.virgin.net /jeff.rollason/newsletter/2005_03_plugin_minimax.htm (1606 words)

	Minimax Enhancements
		One possible approach for designing a minimax cache for Quarto is to use the efficient representation proposed in Section 5.
		Perhaps the most simple and useful extension of the general Minimax algorithm is to limit the depth of the search.
		A generic machine learning algorithm applied to a heuristic function accepts features that are deemed important by the programmer, and assigns weights to them based on its experience.
	zoo.cs.yale.edu /classes/cs490/00-01b/kerner.matthew.mmk29/quarto/node8.html (1151 words)

	Search algorithm - Wikipedia, the free encyclopedia
		In computer science, a search algorithm, broadly speaking, is an algorithm that takes a problem as input and returns a solution to the problem, usually after evaluating a number of possible solutions.
		Brute-force search or "naïve"/uninformed search algorithms use the simplest, most intuitive method of searching through the search space, whereas informed search algorithms use heuristics to apply knowledge about the structure of the search space to try to reduce the amount of time spent searching.
		Minimax which can be highly optimized using alpha-beta pruning is an algorithm to search for good moves in zero-sum games
	en.wikipedia.org /wiki/Informed_search_algorithm (1159 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		The minimax algorithm is presented with a point (B, P) and it returns the point (B’, P’s opponent) such that B’ reflects the “optimal” move that P could have possibly made.
		It is actually possible to compute the minimax value without searching the entire game tree up to the cutoff depth using a method known as alpha-beta pruning.
		When the minimax value function recursively calls itself, instead of initializing the alpha and beta parameters to 0 and 255 respectively, it passes in its own alpha and beta values.
	comet.columbia.edu /msl/2000class/connect4/Software.htm (2852 words)

	CMPSCI 383
		As for the genetic algorithm approach, we had much more success in identifying which words used could best be used to identify the author who used those words.
		Because of all this critical information, I created a second (advanced) minimax algorithm where corner squares are given addition value and adjacent squares a punishing value unless the corner is already taken.
		In addition to the two minimax algorithms, I have created three more, weaker algorithms in order to compare minimax with other ways of finding moves for the computer to play.
	anytime.cs.umass.edu /~bern/teaching/cs383/Honors.html (1602 words)

	Minimax and Alpha-Beta Template
		The minimax algorithm is a recursive algorithm that takes as arguments the current layout of the game board, the depth being searched, and the maximum depth to be searched.
		The algorithm has a two part base case and then calls minimax on the boards resulting from each possible move at the current level.
		Should there be moves for the current player and the maximum depth hasn't yet been searched, minimax generates possible moves at that level, and applies them in turn to the current board, calling minimax on the resulting board.
	www.pressibus.org /ataxx/autre/minimax/node2.html (1145 words)

	Nim, Gametrees, and Minimax
		Minimax is a search method for game trees that provides guidance for good play.
		The minimax technique is then applied to these values as if they were leaf nodes.
		Alpha-Beta pruning is a minimax search modification that speeds of solution searching by truncating searches when they cannot possibly lead to the best solution.
	www.delphiforfun.org /Programs/NIM_Minimax.htm (1651 words)

	java.net: Java Tech: An Intelligent Nim Computer Game, Part 1
		The minimax algorithm determines a player's optimal move by assigning a number to each node that is an immediate child of the player's node.
		With each level that minimax moves up, it alternately determines the maximum or minimum prior to the assignment (which is how minimax gets its name).
		Minimax assigns that value as the minimum to the parent Player B node.
	today.java.net /pub/a/today/2004/05/18/nim1.html (2229 words)

	Remez Algorithm -- from Wolfram MathWorld
		The Remez algorithm (Remez 1934), also called the Remez exchange algorithm, is an application of the Chebyshev alternation theorem that constructs the polynomial of best approximation to certain functions under a number of conditions.
		The Remez algorithm in effect goes a step beyond the minimax approximation algorithm to give a slightly finer solution to an approximation problem.
		Experience has shown that the algorithm converges quickly, and is widely used in practice to design filters with optimal response for a given number of taps.
	mathworld.wolfram.com /RemezAlgorithm.html (383 words)

	"An Enjoyable Game" How HAL Plays Chess, Section 03
		The Minimax Algorithm is a searching algorithm used for the AI in games like chess or tic-tac-toe, which Aasumes that each player always makes the best available move, and determines the best move for the computer.
		The minimax algorithm can be thought of as consisting of two parts: an evaluation function and the minimax rule.
		In theory, the minimax algorithm allows one to play "perfect" chess; that is, the player always makes a winning move in a won position or a drawing move in a drawn position.
	mitpress.mit.edu /e-books/Hal/chap5/five3.html (1018 words)

	Pawn Captures Wyvern: How Computer Chess Can Improve Your Pathfinding
		Once all successors have been examined, the minimax value for that position has been computed and stored in gamma, which can be returned to a higher level within the tree (please refer to Listing 1).
		algorithm [Knuth 1975] improves on the typical minimax algorithm by passing down bounds throughout the tree and can prune off branches that can be shown to have no relevance on the minimax value of the game tree.
		In the minimax algorithm given in Listing 1, all of the information about a node can be accumulated including the best score, the best move from that position, the depth it was searched to.
	www.gamasutra.com /features/20000626/brockington_pfv.htm (5554 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		The special features of the suggested algorithm, however, are that the search directions always decrease the maximum function and attempt to enforce the characterisation of a solution at the same time.
		The Step 3 of the model algorithm is the major part in which the characterisation of the solution is exploited.
		The algorithm presented is a globally convergent algorithm with superlinear convergence rate [23].
	historical.ncstrl.org /tr/temp/ocr/cs-tr.cs.cornell.edu/TR90-1104 (3772 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Our goal is to teach the Z80 microprocessor to play Connect Four competitively against a human opponent by applying the minimax algorithm with alpha-beta pruning.
		The minimax algorithm returns the best possible move, that is, the move that leads to the outcome with the best utility, under the assumption that the opponent plays to minimize the Z80's utility.
		Ideally, the algorithm goes through the entire game tree, all the way to the leaves, to determine the backed-up value of a state.
	comet.columbia.edu /msl/2000class/connect4/Proposal.htm (648 words)

	The Last-Step Minimax Algorithm (Site not responding. Last check: 2007-11-06)
		We develop an algorithm called the Last-step Minimax Algorithm that predicts with the minimax optimal parameter assuming that the current trial is the last one.
		For one-dimensional exponential families, we give an explicit form of the prediction of the Last-step Minimax Algorithm and show that its regret is O(ln T), where T is the number of trials.
		In particular, for Bernoulli density estimation the Last-step Minimax Algorithm is slightly better than the standard Krichevsky-Trofimov probability estimator.
	www-alg.ist.hokudai.ac.jp /~thomas/ALT00/ABS/tw.html (143 words)

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