Editing Minimax (section)

=== Minimax algorithm with alternate moves ===<!-- This section is linked from [[alpha–beta pruning]]. -->
A '''minimax algorithm'''<ref>{{Cite book
| first1 = Stuart J. | last1 = Russell | author1-link = Stuart J. Russell
| first2 = Peter.    | last2 = Norvig  | author2-link = Peter Norvig
| title=[[Artificial Intelligence: A Modern Approach]]
| year = 2021
| edition = 4th 
| isbn = 9780134610993 
| lccn = 20190474
| publisher = Pearson | location = Hoboken
| pages = 149–150
}}
</ref> is a recursive [[algorithm]] for choosing the next move in an n-player [[game theory|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 [[evaluation function|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'''<nowiki/>'s turn to move, '''A''' gives a value to each of their legal moves.

A possible allocation method consists in assigning a certain win for '''A''' as +1 and for '''B''' as −1. This leads to [[combinatorial game theory]] as developed by [[John Horton Conway|John H. 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 maximum of the values resulting from each of '''B'''<nowiki/>'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 (board game)|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 "look-ahead", measured in "[[Ply (chess)|plies]]". For example, the chess computer [[IBM Deep Blue|Deep Blue]] (the first one to beat a reigning world champion, [[Garry Kasparov]] at that time) looked ahead at least 12&nbsp;plies, then applied a heuristic evaluation function.<ref>
{{cite journal 
 | last = Hsu | first = Feng-Hsiung
 | year = 1999
 | title = IBM's Deep Blue chess grandmaster chips
 | journal = IEEE Micro
 | volume = 19  | issue = 2  | pages = 70–81
 | location = Los Alamitos, CA, USA
 | publisher = IEEE Computer Society
 | doi = 10.1109/40.755469
 | quote = During the 1997 match, the software search extended the search to about 40&nbsp;plies along the forcing lines, even though the non-extended search reached only about 12&nbsp;plies.
}}
</ref>

The algorithm can be thought of as exploring the [[node (computer science)|node]]s of a ''[[game tree]]''. The ''effective [[branching factor]]'' of the tree is the average number of [[child node|children]] of each node (i.e., the average number of legal moves in a position).  The number of nodes to be explored usually [[exponential growth|increases exponentially]] with the number of plies (it is less than exponential if evaluating [[forced move]]s 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 [[Computational complexity theory#Intractability|impractical]] to completely analyze games such as chess using the minimax algorithm.

The performance of the naïve minimax algorithm may be improved dramatically, without affecting the result, by the use of [[alpha–beta pruning]]. Other heuristic pruning methods can also be used, but not all of them are guaranteed to give the same result as the unpruned search.

A naïve minimax algorithm may be trivially modified to additionally return an entire [[Variation (game tree)#Principal variation|Principal Variation]] along with a minimax score.