4 execution takes forever even with alpha-beta pruning and move sorting by pieces importance. 6. If it is likely that the opponent (who is always trying to minimize utility) will make all the moves that will lead us to said future point, we should pass that particular future utility value up the tree to represent our current decision node. Hands-on real-world examples, research, tutorials, and cutting-edge techniques delivered Monday to Thursday. I'm new to minimax algorithm, but i understand it's entire concepts as it's easy, my biggest issue is understanding it's implementation to my chess game, no internet solution answers this question. In computer science, a tree refers to a nested data structure in which we start with one “root” node (Level 0), and branching from this root node, we can have any number of “child” nodes (Level 1). This would call Minimize on each child of the board, which calls Maximize on each grandchild, and so on and so forth…. This utility value is then “passed up” to the parent node where it is compared to the current utility value at the parent (if assigned yet). In doing so, the computer assumes that the opponent always selects the best move, minimizing the utility for the computer. In plain English this reads: “For each row, column, and diagonal, if we have 3 X’s in-a-row, we have won the game and should assign a utility of 1.0. The key to the Minimax algorithm is a back and forth between the two players, where the player whose "turn it is" desires to pick the move with the maximum score. If the parent node is in a Maximize cycle, the terminal node utility value replaces the utility at the parent node if it is greater than the current value at the parent node or the parent node has yet to be assigned a utility value. We will use the minimax algorithm for this, and I highly recommend reading up on the Wikipedia article to better understand this decision strategy. This article will take a brief look at how a computer decides its next move using the Minimax Algorithm, but first we need to define a few things: In computer science, a tree refers to a nested data structure in which we start with one “root” node (Level 0), and branching from this root node, we can have any number of “child” nodes (Level 1). Chess engine programmed in Python. Programmed using C++ and rendered with the OpenGL library. This repo includes my solution of the given homework(1/2) in the scope of the Artifical Intelligence(CENG461) course which is given as a technical elective in 2019-2020 Fall semester by Computer Engineering Department at … This article, however, will focus on one general decision-making strategy used by computers across a variety of strategic games, including chess, checkers, mancala, tic-tac-toe, you name it. 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. The Theory of Play and Integral Equations with Skew Symmetric Kernels, Cybernetics or Control and Communication in the Animal and the Machine, La théorie du jeu et les équations intégrales à noyau symétrique, An analog of the minimax theorem for vector payoffs, Experiments With a Multipurpose, Theorem-Proving Heuristic Program, Experiments with the M & N Tree-Searching Program, Evolving Neural Networks to focus Minimax Search, A Survey on Minimax Trees and Associated Algorithms, Interest Search - Another way to do Minimax, The evaluation value and value returned by minimax search, Analog voltage maximizer and minimizer circuits, Little Machine Constructed by Minimax Dadamax in Person from Wikipedia, https://www.chessprogramming.org/index.php?title=Minimax&oldid=20198, Creative Commons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0). 1990s Pressure-sensory chess computer with LCD screen Computer chess is computer architecture encompassing hardware and software capable of playing chess autonomously without human guidance. My main goal is to speed up the time taken to get the best move. Aus diesem Grund wird in der ersten Phase des Algorithmus der schlimmste Zug berechnet, der dann in allen Fällen das Minimum ist (-8, 0, 3). Chess is a game of perfect information — by looking at the board it’s possible to know exactly what an opponent is capable of. Minimax. Applying the above logic to a chess match might be enough to make your head spin, which is why the problem has taken decades and decades to (arguably) solve. Intuitively, we can see … Now that we have an evaluation algorithm, we can start making intelligent decisions! The Minimax algorithm is used to drive the AI of the game. In this algorithm two players play the game, one is called MAX and other is called MIN. It will protect valuable pieces from being captured, and if it looks far enough ahead, it can start to formulate a strategy. The human player should be playing optimally, or actually trying to win. For minimax algorithm to work, the game needs a tree of all* possible moves for the computer to decide which route through a tree to take. This algorithm finds the best move for an AI on a two-player, combinatorial game state on games such as Checkers, Chess or Othello. Minimax is a search algorithm that finds the next optimal move by minimizing the potential loss in a worst case scenario. It concludes that although John von Neumann is usually associated with that concept (1928) , primacy probably belongs to Émile Borel. Chess is a sophisticated game with rather complex utility measures and a vast search space. It keeps playing ahead until it reaches a terminal arrangement of the board terminal state) resulting in a tie, a win, or a loss. At the surface, we might be able to identify what looks like a strong move; however, if this strong move results in an even stronger move by the opponent (shooting our utility down), then was the original move that we made really strong? Apply minimax algorithm with corresponding depth and position values to evaluate the position and decide the best move. visual-studio chess-engine chess ai csharp alpha-beta-pruning chess-game minimax-algorithm chess-ai Updated Apr 23, 2020; C#; fahad462 / Tic_Tac_Toe-Minimax- Star 0 Code Issues Pull requests A simple Tic Tac Toe Game based on MiniMax Theorem In single player or versus computer mode you have less than 1 % chance to win Have doubts Be my guest try it and good luck with that . Ends after 80 moves ( 40 moves from each player ) also chooses the that. One move, utility is calculated using the below utility rule is mostly used for game playing in AI can. A certain depth, which calls Maximize on each child of the game only! We ’ ll improve in the details for now a matter of,. These are what allow the computer chess problem alone, and so forth… called MAX and other is called.. 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Der Minimax-Algorithmus ist ein Algorithmus zur Ermittlung der optimalen Spielstrategie für endliche mit. Add AI to to the python program to make decisions in game theory the zero-sum property of chess minimax. Strategies varying in complexity have been put to the minimax algorithm works well for smaller state space or. Vividh Bharti Announcers Photos With Name, Marriott Houston Medical Center Phone Number, Shock Treatment Movie, Cloverdale Barn Wedding, Maltipoo Puppies For Sale Craigslist, How To Spot Fake Adidas Pants, Akins High School Calendar 2020-2021, Apartments Under \$1,300 In San Jose, Ca, " />
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# minimax algorithm chess

Minimax Algorithm, is a solution to reduce the burden on hardware in chess engine. The original minimax as defined by Von Neumann is based on exact values from game-terminal positions, whereas the minimax search suggested by Norbert Wiener [5] is based on heuristic evaluations from positions a few moves distant, and far from the end of the game. The converse would be true if the parent node were in the Minimize cycle. The minimax algorithm is one of the oldest artificial intelligence algorithms ever. It is widely used in two player turn-based games such as Tic-Tac-Toe, Backgammon, Mancala, Chess, etc. One possible way to decide which move to make next is to simply calculate the utility of each possible next move and select the move with the highest utility. This algorithm is useful in decision-making AI, which is used in popular game engines, like Stockfish for Chess. However, a more in-depth method is needed to further increase the search algorithm. MTD(f) is a new minimax search algorithm, simpler and more efficient than previous algorithms.In tests with a number of tournament game playing programs for chess, checkers and Othello it performed better, on average, than NegaScout/PVS (the AlphaBeta variant used in practically all good chess, checkers, and Othello programs). But as history shows time and time again, this general strategy still works better than human brainpower alone, provided our utility rules are effective. Minimax: The Basic Search Algorithm •Minimax: Assume that both White and Black plays the best moves. Jaap van den Herik's thesis (1983) contains a detailed account of the known publications on that topic. We can see that it’s X’s turn, and there are only 3 possible moves, and hence, 3 child nodes. This means that on average a game of Chess ends after 80 moves (40 moves from each player). In this video, we go over the Minimax algorithm with Alpha Beta pruning that we will use in the following videos. The Minimax algorithm works well for smaller state space, or in other words, smaller board games. Chess Simulation. With minimax in place, our algorithm is starting to understand some basic tactics of chess: Minimax with depth level 2. A detailed explanation isavailable on Wikipedia, but here is my quick, less rigorous outline: 1. Minimax is a recursive algorithm which is used to choose an optimal move for a player assuming that the other player is also playing optimally. Secondarily, the algorithm moves horizontally, or among other sibling nodes . It construct the tree like above and applies Minimax Algorithm. This means that all of our children, or the children of a particular node, will be our opponent’s move. The computer is essentially applying the following logic: Your next move is only as strong as the opponent’s following move is weak. It concludes that although John von Neumann is usually associated with that concept (1928) [3] , primacy probably belongs to Émile Borel. We will recap … The only problem is the algorithm takes a long time. The figure below displays a tic-tac-toe board midway through the game with a very simple (probably not optimal) utility rule. You're not doing quiescence search, so the dumb moves are likely due to the well-known horizon effect that fixed depth minimax searches are susceptible to. For each possible move, utility is calculated using the below utility rule. The Minimax algorithm is the core of several game-playing AI for making decisions on the best move. In board games, we can think of the root node (Level 0) as the current state of the board, and the nodes at Level 1 as each possible state of the board depending on what the next move will be. Get the FEN of the current position Since we are performing game playing, we will take turns, just like in a game of chess or tic-tac-toe; we take a turn, and then our opponent takes a turn. Intuitively, we might be able to think about how this cycle occurs recursively over and over until we are able to populate the next move nodes (Level 1) with utility values. We can see that the winning board has the maximum utility, so our utility rule isn’t totally useless, although it would certainly benefit from a “defensive” component, which would consider how close O is to a win. We will use the minimax algorithm for this, and I highly recommend reading up on the Wikipedia article to better understand this decision strategy. ). So if you are going through a similar journey, I would like to introduce you to the Minimax algorithm. Hence the game-tree complexity of the board game is 3580≈10123, Yet, if we consider only the sensible moves (non stupid moves), the state-space complexit… Minimax is an algorithm we can use to evaluate all possible moves a player can make. The expansive timeframe over which the chess computer problem has been pondered lends credence to the complexity of the solution. In der Regel, aber nicht aussc… Get the FEN of the current position For more complex scenarios, including games like chess and go, minimax would, at the very least, have to be combined with other techniques. The Minimax algorithm is the core of several game-playing AI for making decisions on the best move. So Level 1 contains nodes representing Opponent A’s possible moves, and Level 2 contains nodes representing Opponent B’s possible moves for each and every possible move Opponent A can make back in Level 1. Our evaluation will be relative to a player, and that player will be trying to get the best score possible (maximize their position). This is in stark contrast to a “breadth-first search”, which does just the opposite — it primarily moves horizontally among fellow sibling nodes, searching a whole “level” at a time, and secondarily works its way down the tree. 5. We can represent chessboard positions as … At this point, the chess AI is starting to make good moves. Send the FEN to the python program to make that move. In Minimax the two players are called maximizer and minimizer. But the great minds behind the chess computer problem had started publishing in the subject nearly 6 decades earlier. f) A Minimax Algorithm faster than NegaScout. It is shown that with e ective move ordering and probing the Star2 algorithm considerably outperforms Expectimax. In other words, the maximizer works to get the highest score, while the minimizer tries get the lowest score by tr… Get the FEN of the current position Instead of using two separate subroutines for the Min player and the Max player, it passes on the negated score due to following mathematical relation: max (a, b) == -min (-a, -b) Firstly, an evaluation function f: P → R f:\mathbb{P} \rightarrow \mathbb{R} f: P → R from the set of positions to real numbers is required, representing the payoff to the first player. It can incorporate a large variety of factors and weigh them as the programmer sees fit. Alpha–beta pruning is a search algorithm that seeks to decrease the number of nodes that are evaluated by the minimax algorithm in its search tree. Each time your opponent takes a turn, the worst move for you is chosen (min), as it benefits your opponent the most 4. As it turns out, computers can do this much more efficiently than even the best of the best chess masters out there. The best chess programming approach combined the "minimax" algorithm with the "alpha-beta pruning" technique. Star2 allows strong backgammon programs to conduct depth 5 full-width searches (up from 3) under tournament conditions on regular hardware without using risky forward … But the underlying principle of what goes into programming a chess-playing computer is largely the same as shown in the simplified example above — look at possible moves in the future, decide how good these moves are, and predict whether the opponent will make all the right moves to lead you there. Depth limits are set for games involving complex search spaces, in which it would not be feasible to search the entire network of possible moves within a reasonable amount of time. Below the pseudo code for an indirect recursive depth-first search. Enter the analysis mode for a starting chess position. The above schematic is oversimplified in the sense that an opponent only has 3 possible moves any given turn. Of course this is not a safe assumption, but lo and behold, it tends to work out pretty well regardless. Make this move in the python program. board state evaluation based on pieces weights (pretty base solution) minimax search algorithm for best move/optional depth; alpha-beta search tree pruning; game class for games with different players; game result stats; Problems Known as the father of modern computer science, Alan Turing is credited with provoking the investigation dating all the way back to the 1940’s. My chess game is over, everything has been finished, except for some special (like en passant)moves. Make learning your daily ritual. Let’s Find Out, 4 Machine Learning Concepts I Wish I Knew When I Built My First Model, 7 Beginner to Intermediate SQL Interview Questions for Data Analytics roles. Minimax is a kind of backtracking algorithm that is used in decision making and game theory to find the optimal move for a player, assuming that your opponent also plays optimally. But what is it that we are actually doing, and how does this help a computer make a decision? I have recently found out about the minimax algorithm and how it interacts with Tic Tac Toe. In turn, the scores for each of the available moves are determined by the opposing player deciding which of … I've decided to continue my adventure after minimax implementation for tic tac toe. This means that the evaluation of a position is equivalent to the negation of the evaluation from the opponent's viewpoint. The branching factor corresponds to the average number of moves a player can execute during its turn. This is often times the strategy of the average human when it comes to board games, and certainly, games can be won this way. ⭐https://chessprogramming.wikispaces.com/Chess Code Repository :⭐https://github.com/amir650/BlackWidow-ChessIDE:⭐https://www.jetbrains.com/idea/Reading … It is an adversarial search algorithm used commonly for machine playing of two-player games (Tic-tac-toe, Chess, Go, etc. Using minimax, the computer searched a game tree of possible moves and counter-moves, evaluating the best move on its turn and the worst move on its opponent’s turn. 5. Every strong engine before AlphaZero- such as Stockfish, Rybka, Crafty, and Fritz- was an AB engine. chess2.h Minimax is a type of backtracking algorithm. How they work: the Minimax algorithm Chess is a so-called “zero-sum game”, which simply means a game in which if one player wins the other loses. 3. These are what allow the computer to make a decision. Let’s say it’s the computer’s turn. *All possible moves in this game because it is short, but in such games like standard chess it is not possible because of the huge scaling – 10^120 possible variations , so most of the times tree is being generated dynamically. It is applicable only to 2-player games, where the opponent’s move is predictable beforehand. Let’s take tic-tac-toe, for example, which can have relatively simple utility measures. Such as Chess, Checkers, tic-tac-toe, go, and various tow-players game. a common way of implementing minimax and derived algorithms. The algorithm incorporates three basic functions: Maximize and Minimize, as well as a Utility Calculation function. It’s maximizer and minimizer. Minimax Algorithm and Implementation. In this article, we will focus on implementing some classic algorithms. Min-Max algorithm is mostly used for game playing in AI. Often times, in chess for instance, the number of possible moves can be much, much greater, causing our game tree to become complicated in a hurry. Next, we’ll put the pieces together to see how a computer “thinks” about its next move. 3. “chess pieces on board” by Felix Mittermeier on Unsplash. Make this move in the python program. Schließlich wird das Maximum berechnet (3). In this post, I assume that the reader is familiar with the algorithm and its inherent code size due to its frequent implementation with three distinct functions. … For eg., for a Tic-Tac-Toe game, there are 3 9 =196839 possible board combinations which can be easily calculated. Simple Chess AI with Minimax Algorithm Introduction. This is especially true when it comes to games like chess, where variations of the Minimax algorithm are what is used to build the strongest chess-playing programs in existence. Before diving in, we will make 2 assumptions about our game: The premise of the algorithm is that the computer will calculate its next best move by evaluating the utility of the board several turns down the road. For minimax algorithm to work, the game needs a tree of all* possible moves for the computer to decide which route through a tree to take. Alpha–beta pruning is a search algorithm that seeks to decrease the number of nodes that are evaluated by the minimax algorithm in its search tree.It is an adversarial search algorithm used commonly for machine playing of two-player games (Tic-tac-toe, Chess, Go, etc. someone wins the game) or a pre-determined depth limit. In other words, Level 1 nodes account for each and every possible next move. Enter the analysis mode for a starting chess position. The move with the best evaluation is chosen. Apply minimax algorithm with corresponding depth and position values to evaluate the position and decide the best move. The Minimax a… Here, we will implement on how to add AI to to the game. In combinatorial games such as chess and Go, the minimax algorithm gives a method of selecting the next optimal move. This page was last edited on 14 July 2020, at 13:47. Here are the values for each piece on the board, which is represented by an 8x8 integer array. Take a look, 6 Data Science Certificates To Level Up Your Career, Stop Using Print to Debug in Python. How utility is calculated is entirely up to the programmer. Minimax is a type of backtracking algorithm. It’s worth emphasizing that minimax works fine for a toy scenario like tic-tac-toe: There are few distinct game positions - 765 if we take rotation and reflection symmetry into account. This Algorithm computes the minimax decision for the current state. Minimax is a simple algorithm that tells you which move to play in a game. Implemented. Further there is a conceivable claim that the first to credit should go to Charles Babbage . Utility can be thought of as a way to “score” each possible move based on its potential to result in a win. lard’s *-Minimax algorithms applied to a real{world domain: backgam-mon. Apply minimax algorithm with corresponding depth and position values to evaluate the position and decide the best move. At a minimum you should extend search for any forced moves, checks or captures where a piece captures one of equal or greater value. We follow each move to a certain depth, which in this instance is how many moves ahead we want to look. Textbooks have been written on the computer chess problem alone, and many strategies varying in complexity have been put to the test. Bonus Optimisation. For eg. King = 10; Pawn = 1; Bishop = 3; Knight = 2; Queen = 6; Rook = 5; negative values of the same represent black pieces. This is the line of thinking behind the algorithm, applied over several levels of our tree. But what differentiates the masters from the ordinary is the ability to think several moves ahead. For instance, number of blank spaces on the board, the location of the opponent’s current pieces, the location of our current pieces, how close we are to a winning formation, etc. This general strategy is encased in a widely-used algorithm in gaming theory, called the Minimax Algorithm. If player A can win in one move, their best move is that winning move. Possible solution: tree caching, better heuristic, parallelization (oh that's hard), build tree only for some promising moves (for example in some range from the opponent). Minimax algorithm takes into consideration that the opponent is also playing optimally, which makes it useful for two-player games such as checker, chess, Tic-tac-toe, go and many others. Minimax is a decision rule which simulate the decision of a player to find the optimal move for the player. Chess is a game of perfect information — by looking at the board it’s possible to know exactly what an opponent is capable of. The clip below might help in visualizing this concept. Here is my chess class to that holds everything. Please read my post on Minimax algorithm if you haven’t already.. Alpha-beta pruning is based on the Branch and bound algorithm design paradigm, where we will generate uppermost and lowermost possible values to our optimal solution and using them, discard any … It construct the tree like above and applies Minimax Algorithm. Question about minimax and chess. Send the FEN to the python program to make that move. I have problem with my own Chess Engine using minimax algorithm to search for chess moves I use a 5 plies depth search and with only material/bonus/mobility evaluation , but it also make dumb moves and sacrifices valuable pieces even when I give to them infinity (which is sure a search problem), I'm not using any types of pruning and gives a 5 depth search result in few seconds. Making moves at random or trying to lose might actually interfere with the algorithm effectiveness. Like a professional chess player, this algorithm sees a few steps ahead and puts itself in the shoes of its opponent. This Algorithm computes the minimax decision for the current state. Chess is a two-player strategy board game played on checkered board with 64 squares arranged in an 8x8 grid. So if you are going through a similar journey, I would like to introduce you to the Minimax algorithm. Further there is a conceivable claim that the first to credit should go to Charles Babbage [4]. black or white players comprises of several pieces, so what piece is the AI predicting moves for, as the player could move any of their 16 pieces? As a consequence, total wins minus total losses equals zero, from where “zero-sum” is derived. Minimax Now that we have an evaluation algorithm, we can start making intelligent decisions! , where each player can win, lose, or draw. This is something we’ll improve in the following step. If you want to write a program that is able to play a strategy game, there are good chances that you will be looking at a Minimax algorithm. Game Tree We can represent chessboard positions as nodes in a *game tree. Make this move in the python program. It assumes that the opponent will play with optimal choice too. Minimax for depth > 4 execution takes forever even with alpha-beta pruning and move sorting by pieces importance. 6. If it is likely that the opponent (who is always trying to minimize utility) will make all the moves that will lead us to said future point, we should pass that particular future utility value up the tree to represent our current decision node. Hands-on real-world examples, research, tutorials, and cutting-edge techniques delivered Monday to Thursday. I'm new to minimax algorithm, but i understand it's entire concepts as it's easy, my biggest issue is understanding it's implementation to my chess game, no internet solution answers this question. In computer science, a tree refers to a nested data structure in which we start with one “root” node (Level 0), and branching from this root node, we can have any number of “child” nodes (Level 1). This would call Minimize on each child of the board, which calls Maximize on each grandchild, and so on and so forth…. This utility value is then “passed up” to the parent node where it is compared to the current utility value at the parent (if assigned yet). In doing so, the computer assumes that the opponent always selects the best move, minimizing the utility for the computer. In plain English this reads: “For each row, column, and diagonal, if we have 3 X’s in-a-row, we have won the game and should assign a utility of 1.0. The key to the Minimax algorithm is a back and forth between the two players, where the player whose "turn it is" desires to pick the move with the maximum score. If the parent node is in a Maximize cycle, the terminal node utility value replaces the utility at the parent node if it is greater than the current value at the parent node or the parent node has yet to be assigned a utility value. We will use the minimax algorithm for this, and I highly recommend reading up on the Wikipedia article to better understand this decision strategy. This article will take a brief look at how a computer decides its next move using the Minimax Algorithm, but first we need to define a few things: In computer science, a tree refers to a nested data structure in which we start with one “root” node (Level 0), and branching from this root node, we can have any number of “child” nodes (Level 1). Chess engine programmed in Python. Programmed using C++ and rendered with the OpenGL library. This repo includes my solution of the given homework(1/2) in the scope of the Artifical Intelligence(CENG461) course which is given as a technical elective in 2019-2020 Fall semester by Computer Engineering Department at … This article, however, will focus on one general decision-making strategy used by computers across a variety of strategic games, including chess, checkers, mancala, tic-tac-toe, you name it. 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. The Theory of Play and Integral Equations with Skew Symmetric Kernels, Cybernetics or Control and Communication in the Animal and the Machine, La théorie du jeu et les équations intégrales à noyau symétrique, An analog of the minimax theorem for vector payoffs, Experiments With a Multipurpose, Theorem-Proving Heuristic Program, Experiments with the M & N Tree-Searching Program, Evolving Neural Networks to focus Minimax Search, A Survey on Minimax Trees and Associated Algorithms, Interest Search - Another way to do Minimax, The evaluation value and value returned by minimax search, Analog voltage maximizer and minimizer circuits, Little Machine Constructed by Minimax Dadamax in Person from Wikipedia, https://www.chessprogramming.org/index.php?title=Minimax&oldid=20198, Creative Commons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0). 1990s Pressure-sensory chess computer with LCD screen Computer chess is computer architecture encompassing hardware and software capable of playing chess autonomously without human guidance. My main goal is to speed up the time taken to get the best move. Aus diesem Grund wird in der ersten Phase des Algorithmus der schlimmste Zug berechnet, der dann in allen Fällen das Minimum ist (-8, 0, 3). Chess is a game of perfect information — by looking at the board it’s possible to know exactly what an opponent is capable of. Minimax. Applying the above logic to a chess match might be enough to make your head spin, which is why the problem has taken decades and decades to (arguably) solve. Intuitively, we can see … Now that we have an evaluation algorithm, we can start making intelligent decisions! The Minimax algorithm is used to drive the AI of the game. In this algorithm two players play the game, one is called MAX and other is called MIN. It will protect valuable pieces from being captured, and if it looks far enough ahead, it can start to formulate a strategy. The human player should be playing optimally, or actually trying to win. For minimax algorithm to work, the game needs a tree of all* possible moves for the computer to decide which route through a tree to take. This algorithm finds the best move for an AI on a two-player, combinatorial game state on games such as Checkers, Chess or Othello. Minimax is a search algorithm that finds the next optimal move by minimizing the potential loss in a worst case scenario. It concludes that although John von Neumann is usually associated with that concept (1928) , primacy probably belongs to Émile Borel. Chess is a sophisticated game with rather complex utility measures and a vast search space. It keeps playing ahead until it reaches a terminal arrangement of the board terminal state) resulting in a tie, a win, or a loss. At the surface, we might be able to identify what looks like a strong move; however, if this strong move results in an even stronger move by the opponent (shooting our utility down), then was the original move that we made really strong? 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