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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 the player's actions; equivalently, it is the largest value the player can be sure to get when they know the actions of the other players. Its formal definition is:[3]

Frequently, in game theory, maximin is distinct from minimax. 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.

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A simple version of the minimax algorithm, stated below, deals with games such as tic-tac-toe, where each player can win, lose, or draw. If player A can win in one move, their 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 "best" move is. 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.e., to maximize B's own chances of winning).

A minimax algorithm[5] is a recursive algorithm for choosing the next move in an n-player game, usually a two-player game. 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 their legal moves.

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 "look-ahead", measured in "plies". 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 12 plies, then applied a heuristic evaluation function.[6]

The algorithm can be thought of as exploring the nodes of a game tree. 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 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 minimax function returns a heuristic value for leaf nodes (terminal nodes and nodes at the maximum search depth). Non-leaf nodes inherit their 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.

A key feature of minimax decision making is being non-probabilistic: in contrast to decisions using expected value or expected utility, it makes no assumptions about the probabilities of various outcomes, just scenario analysis of what the possible outcomes are. It is thus robust to changes in the assumptions, in contrast to these other decision techniques. Various extensions of this non-probabilistic approach exist, notably minimax regret and Info-gap decision theory.

The concept of "lesser evil" voting (LEV) can be seen as a form of the minimax strategy where voters, when faced with two or more candidates, choose the one they perceive as the least harmful or the "lesser evil." To do so, "voting should not be viewed as a form of personal self-expression or moral judgement directed in retaliation towards major party candidates who fail to reflect our values, or of a corrupt system designed to limit choices to those acceptable to corporate elites" rather as an opportunity to reduce harm or loss.[7]

I think what you're missing here is how minimax works. Minimax enumerates all of the possibilities to a specified depth D, then assigns a score to the nodes (game states) at D, and moving back up the tree, returns the MAX or MIN node at each depth based on whether I'm the maximizing player or the minimizing player.

The book provides an introduction to minimax methods in critical point theory and shows their use in existence questions for nonlinear differential equations. An expanded version of the author's 1984 CBMS lectures, this volume is the first monograph devoted solely to these topics. Among the abstract questions considered are the following: the mountain pass and saddle point theorems, multiple critical points for functionals invariant under a group of symmetries, perturbations from symmetry, and variational methods in bifurcation theory.

So I'm following a CodeHS course for building an AI tic tac toe game, however while I was trying to finish up my minimax function, it returns wrong answers for the 2nd board test(the tests at the bottom) I don't know why. Help me out please!

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In game theory, minimax is a decision rule used to minimize the worst-case potential loss; in other words, a player considers all of the best opponent responses to his strategies, and selects the strategy such that the opponent's best strategy gives a payoff as large as possible.

The name "minimax" comes from minimizing the loss involved when the opponent selects the strategy that gives maximum loss, and is useful in analyzing the first player's decisions both when the players move sequentially and when the players move simultaneously. In the latter case, minimax may give a Nash equilibrium of the game if some additional conditions hold.

Minimax is also useful in combinatorial games, in which every position is assigned a payoff. The simplest example is assigning a "1" to a winning position and "-1" to a losing one, but as this is difficult to calculate for all but the simplest games, intermediate evaluations (specifically chosen for the game in question) are generally necessary. In this context, the goal of the first player is to maximize the evaluation of the position, and the goal of the second player is to minimize the evaluation of the position, so the minimax rule applies. This, in essence, is how computers approach games like chess and Go, though various computational improvements are possible to the "naive" implementation of minimax.

Suppose player \(i\) chooses strategy \(s_i\), and the remaining players choose the strategy profile \(s_{-i}\). If \(u_i(S)\) denotes the utility function for player \(i\) on strategy profile \(s\), the minimax of a game is defined as

The minimax choice for the first player is strategy 2, and the minimax choice for the second player is also strategy 2. But both players choosing strategy 2 does not lead to a Nash equilibrium; either player would choose to change their strategy given knowledge of the other's. In fact, the mixed minimax strategies of:

In combinatorial games such as chess and Go, the minimax algorithm gives a method of selecting the next optimal move. Firstly, an evaluation function \(f:\mathbb{P} \rightarrow \mathbb{R}\) from the set of positions to real numbers is required, representing the payoff to the first player. For example, a chess position with evaluation +1.5 is significantly in favor of the first player, while a position with evaluation \(-\infty\) is a chess position in which White is checkmated. Once such a function is known, each player can apply the minimax principle to the tree of possible moves, thus selecting their next move by truncating the tree at some sufficiently deep point.

Of course, games like chess and go are vastly more complicated, as dozens of moves are possible at any possible point (rather than the 1-3 in the above example). Thus it is infeasible to completely solve these games using a minimax algorithm, meaning that the evaluation function is used at a sufficiently deep point in the tree (for example, most modern chess engines apply a depth of somewhere between 16 and 18) and minimax is used to fill in the rest of this relatively small tree. 17dc91bb1f