Minimax - Wikipedia, the free encyclopedia. Minimax (sometimes Min. Max or MM. Originally formulated for two- player zero- sumgame theory, covering both the cases where players take alternate moves and those where they make simultaneous moves, it has also been extended to more complex games and to general decision- making in the presence of uncertainty.
Game theory. Its formal definition is. Then, we determine which action player i. The result of the combination of both moves is expressed in a payoff table: LRT3,1. M5,0- 1. 0,1. B- 1. For the sake of example, we consider only pure strategies. Check each player in turn: If both players play their maximin strategies (T,L), the payoff vector is (3,1). In contrast, the only Nash equilibrium in this game is (B,R), which leads to a payoff vector of (4,4).
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The minimax value of a player is the smallest value that the other players can force the player to receive, without knowing his actions. Equivalently, it is the largest value the player can be sure to get when he knows the actions of the other players. Its formal definition is. Consider the game in the above example: In zero- sum games. The name minimax arises because each player minimizes the maximum payoff possible for the other. See also example of a game without a value. Example. Suppose each player has three choices and consider the payoff matrix for A displayed on the right.
Assume the payoff matrix for B is the same matrix with the signs reversed (i. Then, the minimax choice for A is A2 since the worst possible result is then having to pay 1, while the simple minimax choice for B is B2 since the worst possible result is then no payment. However, this solution is not stable, since if B believes A will choose A2 then B will choose B1 to gain 1; then if A believes B will choose B1 then A will choose A1 to gain 3; and then B will choose B2; and eventually both players will realize the difficulty of making a choice. So a more stable strategy is needed. Some choices are dominated by others and can be eliminated: A will not choose A3 since either A1 or A2 will produce a better result, no matter what B chooses; B will not choose B3 since some mixtures of B1 and B2 will produce a better result, no matter what A chooses. A can avoid having to make an expected payment of more than 1.
Similarly, B can ensure an expected gain of at least 1/3, no matter what A chooses, by using a randomized strategy of choosing B1 with probability 1. These mixed minimax strategies are now stable and cannot be improved. Maximin. Minimax is used in zero- sum games to denote minimizing the opponent's maximum payoff. In a zero- sum game, this is identical to minimizing one's own maximum loss, and to maximizing one's own minimum gain. In non- zero- sum games, this is not generally the same as minimizing the opponent's maximum gain, nor the same as the Nash equilibrium strategy. In repeated games. One of the central theorems in this theory, the folk theorem, relies on the minimax values.
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Minimax (sometimes MinMax or MM) is a decision rule used in decision. Look up minimax in Wiktionary, the free dictionary. Hazewinkel, Michiel, ed.
Combinatorial game theory. If player A can win in one move, his best move is that winning move.
If player B knows that one move will lead to the situation where player A can win in one move, while another move will lead to the situation where player A can, at best, draw, then player B's best move is the one leading to a draw. Late in the game, it's easy to see what the . The Minimax algorithm helps find the best move, by working backwards from the end of the game. At each step it assumes that player A is trying to maximize the chances of A winning, while on the next turn player B is trying to minimize the chances of A winning (i. B's own chances of winning). Minimax algorithm with alternate moves. A value is associated with each position or state of the game.
This value is computed by means of a position evaluation function and it indicates how good it would be for a player to reach that position. The player then makes the move that maximizes the minimum value of the position resulting from the opponent's possible following moves. If it is A's turn to move, A gives a value to each of his legal moves. A possible allocation method consists in assigning a certain win for A as +1 and for B as . This leads to combinatorial game theory as developed by John Horton Conway. An alternative is using a rule that if the result of a move is an immediate win for A it is assigned positive infinity and, if it is an immediate win for B, negative infinity. The value to A of any other move is the minimum of the values resulting from each of B's possible replies.
For this reason, A is called the maximizing player and B is called the minimizing player, hence the name minimax algorithm. The above algorithm will assign a value of positive or negative infinity to any position since the value of every position will be the value of some final winning or losing position. Often this is generally only possible at the very end of complicated games such as chess or go, since it is not computationally feasible to look ahead as far as the completion of the game, except towards the end, and instead positions are given finite values as estimates of the degree of belief that they will lead to a win for one player or another. This can be extended if we can supply a heuristic evaluation function which gives values to non- final game states without considering all possible following complete sequences. We can then limit the minimax algorithm to look only at a certain number of moves ahead. This number is called the .
For example, the chess computer Deep Blue (the first one to beat a reigning world champion, Garry Kasparov at that time) looked ahead at least 1. The effective branching factor of the tree is the average number of children of each node (i. The number of nodes to be explored usually increases exponentially with the number of plies (it is less than exponential if evaluating forced moves or repeated positions).
The number of nodes to be explored for the analysis of a game is therefore approximately the branching factor raised to the power of the number of plies. It is therefore impractical to completely analyze games such as chess using the minimax algorithm.
The performance of the na. Other heuristic pruning methods can also be used, but not all of them are guaranteed to give the same result as the un- pruned search. A na. Non leaf nodes inherit their value, best. Value, from a descendant leaf node. The heuristic value is a score measuring the favorability of the node for the maximizing player. Hence nodes resulting in a favorable outcome, such as a win, for the maximizing player have higher scores than nodes more favorable for the minimizing player. The heuristic value for terminal (game ending) leaf nodes are scores corresponding to win, loss, or draw, for the maximizing player.
For non terminal leaf nodes at the maximum search depth, an evaluation function estimates a heuristic value for the node. The quality of this estimate and the search depth determine the quality and accuracy of the final minimax result. Minimax treats the two players (the maximizing player and the minimizing player) separately in its code. Based on the observation that max(a,b)=.
The algorithm generates the tree on the right, where the circles represent the moves of the player running the algorithm (maximizing player), and squares represent the moves of the opponent (minimizing player). Because of the limitation of computation resources, as explained above, the tree is limited to a look- ahead of 4 moves. The algorithm evaluates each leaf node using a heuristic evaluation function, obtaining the values shown. The moves where the maximizing player wins are assigned with positive infinity, while the moves that lead to a win of the minimizing player are assigned with negative infinity. At level 3, the algorithm will choose, for each node, the smallest of the child node values, and assign it to that same node (e. The next step, in level 2, consists of choosing for each node the largest of the child node values. Once again, the values are assigned to each parent node.
The algorithm continues evaluating the maximum and minimum values of the child nodes alternately until it reaches the root node, where it chooses the move with the largest value (represented in the figure with a blue arrow). This is the move that the player should make in order to minimize the maximum possible loss. Minimax for individual decisions. For example, deciding to prospect for minerals entails a cost which will be wasted if the minerals are not present, but will bring major rewards if they are. One approach is to treat this as a game against nature (see move by nature), and using a similar mindset as Murphy's law or resistentialism, take an approach which minimizes the maximum expected loss, using the same techniques as in the two- person zero- sum games. In addition, expectiminimax trees have been developed, for two- player games in which chance (for example, dice) is a factor. Minimax criterion in statistical decision theory.
We also assume a risk function. R(. An estimator is Bayes if it minimizes the average risk. It is thus robust to changes in the assumptions, as these other decision techniques are not.
Various extensions of this non- probabilistic approach exist, notably minimax regret and Info- gap decision theory. Further, minimax only requires ordinal measurement (that outcomes be compared and ranked), not interval measurements (that outcomes include .
Compare to expected value analysis, whose conclusion is of the form: . The Difference Principle. Rawls defined this principle as the rule which states that social and economic inequalities should be arranged so that . Cambridge University Press.
A Course in Game Theory. Cambridge, MA: MIT, 1. Print.^Russell, Stuart J.; Norvig, Peter (2. Artificial Intelligence: A Modern Approach (2nd ed.), Upper Saddle River, New Jersey: Prentice Hall, pp.