Minimax in chess - definition and use

Minimax

Definition

Minimax is a decision-making principle from game theory that chooses a move which maximizes a player's minimum guaranteed outcome, assuming the opponent will play optimally to minimize that outcome. In zero-sum, perfect-information games like chess, minimax evaluates a game tree by backing up results from the leaves (end positions or depth limits) to the root, alternating between “max” nodes (your turn) and “min” nodes (opponent’s turn).

How it is used in chess

Chess engines and strong human calculation both embody the minimax idea: select moves that hold up against the opponent’s best replies, not just the most cooperative ones.

Game tree search: Engines generate legal moves, explore replies, and propagate values up the tree. White nodes try to maximize the evaluation; Black nodes try to minimize it.
Evaluation function: Leaf nodes (end positions or depth-limited nodes) are scored by an evaluation function (material, king safety, space, pawn structure, etc.). Mates are represented as special large values (e.g., +M3 means mate for the side to move in three moves; -M5 means mate against in five).
Alpha–beta pruning: A speed-up of minimax that prunes branches that cannot affect the final choice. Good move ordering can reduce the number of nodes from roughly b^d towards b^(d/2) in the best case (b = branching factor, d = depth).
Iterative deepening and time management: Engines repeatedly search to increasing depths, using results to order moves better in the next iteration and to ensure a usable result if time expires.
Quiescence search: Extends the search in “tactical” positions (checks, captures, promotions) to avoid the horizon effect—misjudging volatile positions by cutting off too early.
Transposition tables: Cache positions reached via different move orders to avoid redundant search and to reuse best lines.
Negamax formulation: A simplified implementation trick: store positions from the current player’s perspective so minimax becomes “maximize the negated value of the child.”
Modern engines: Classical engines (e.g., Stockfish with NNUE evaluation) are alpha–beta/minimax based. Neural-network engines like Lc0 use Monte Carlo Tree Search, but the backup step still reflects the same “best reply” principle that minimax formalizes.

Strategic and historical significance

Origins: The minimax theorem (John von Neumann, 1928) underpins optimal play in zero-sum games. Claude Shannon’s 1950 paper outlined applying search and evaluation—effectively minimax—to chess.
Deep Blue (1997): IBM’s Deep Blue used alpha–beta pruning with massive parallelism and a rich evaluation function to defeat Garry Kasparov (Kasparov vs. Deep Blue, 1997), a landmark achievement for computer chess built on minimax search.
Tablebases: Endgame tablebases compute perfect minimax results (win/draw/loss and depth to mate) for all positions within material limits. They are “solved” subgames of chess.
Solving chess: To “solve” chess is to know the minimax value of the initial position (win/draw/loss with perfect play). Current evidence strongly indicates a draw, but it is not proven.

Examples

Example 1 — Endgame (opposition and guaranteed outcome):

Position: White king e5, White pawn e6; Black king e7. White to move.
If White “hopes” for cooperation with 1. Kf5?, Black plays 1... Ke8 2. Kf6 Kf8 and draws. Minimax thinking asks: what if Black plays best?
The correct minimax line is 1. Kd5! forcing zugzwang. After 1... Ke8 (only move to hold), 2. Kd6 Kd8 3. e7+ Ke8 4. Ke6 and Black must yield: the pawn will promote. White chose the move that secures a win against best defense, not the move that only works if Black blunders.

Example 2 — Choosing by worst-case score:

Imagine your candidate moves produce these backed-up (minimax) worst-case evaluations from your engine/search:

1. Rc1: worst-case +0.40 (slight advantage even if Black defends best)
1. Qh5: worst-case -0.20 (you get worse if Black finds the key resource)
1. g4: worst-case +0.10

Minimax selects 1. Rc1, the move with the best guaranteed outcome after the opponent’s best replies.

Example 3 — Tactical forcing lines:

In sharp positions, minimax encourages calculating forcing sequences (checks, captures, threats). Suppose a line like 1. Qxh7+! Kxh7 2. Rh3+ Kg8 3. Ne7# appears in your analysis tree. If every Black deviation before mate leads to an even worse evaluation (e.g., losing heavy material or mating sooner), the leaf “mate” scores back up to earlier nodes, making 1. Qxh7+ the minimax choice over quieter alternatives.

Common pitfalls and related concepts

Horizon effect: A looming threat beyond the search depth can cause errors (e.g., “postponing” a material loss). Quiescence search and selective extensions mitigate this.
Move ordering: Good ordering (checks, captures, killer moves, history heuristics) improves alpha–beta pruning efficiency dramatically.
Evaluation traps: If the eval function misjudges structures (e.g., overvalues a passed pawn when blockaded), the backed-up minimax result can be misleading. Strong evals are crucial.
Draw management: In positions where one line yields a safe draw and another yields a speculative attack that could be refuted, minimax favors the line that guarantees the draw if the alternative can be neutralized by best play.

Interesting facts and anecdotes

Shannon number: The estimated size of the chess game tree is about 10^120, which is why pruning (alpha–beta), transposition tables, and good heuristics are essential for minimax to reach practical depths.
Mate scores: Engines annotate forced mates as +M# or -M#. These are absolute in the minimax sense: no defense can change the outcome if the score says +M# for the side to move.
Minimax in human thought: “If I play this, what’s their strongest reply?” is everyday minimax. Coaches often warn against “hope chess,” which ignores the opponent’s best defense.
Leela and backups: Even though Leela Chess Zero uses Monte Carlo Tree Search with a neural net, its node backups still reflect the same principle of choosing lines that withstand the opponent’s best resistance.