Minimax evaluates choices in a two-player model with opposing payoffs by considering each player’s best response at their turn. For a finite deterministic game tree with perfect information, the values can be propagated backward from terminal outcomes. This tutorial uses a four-leaf invented tree to explain the calculation and the limits of the resulting guarantee.
State the model before evaluating the tree
The example has two players, MAX and MIN. MAX prefers a higher terminal score; MIN prefers a lower score, representing opposite payoffs. MAX chooses Left or Right, then MIN observes that choice and chooses one of two terminal branches. The MIT handout on game-tree search distinguishes deterministic, chance and imperfect-information settings. Our example is finite, deterministic and fully observed. Those properties support the elementary calculation below; they should not be silently assumed for every strategic interaction.
Give the leaves an explicit meaning
After Left, MIN can select a terminal score of three or seven for MAX. After Right, MIN can select two or nine. The scores are invented utility labels, not observed performance. A terminal value represents the complete outcome of this model. It differs from a heuristic score assigned to an unfinished position because a search ran out of time. Keep that distinction visible: the backward calculation can be exact for a supplied tree while the supplied leaf values remain approximations to a larger problem.
Work backward from the terminal scores
| MAX branch | MIN terminal option | Score for MAX | Value backed up to branch |
|---|---|---|---|
| Left | First | 3 | min(3,7)=3 |
| Left | Second | 7 | min(3,7)=3 |
| Right | First | 2 | min(2,9)=2 |
| Right | Second | 9 | min(2,9)=2 |
At the root, MAX compares the backed-up values and selects max(3,2)=3. The largest leaf, nine, does not determine the root choice because MIN controls whether that leaf is reached. The calculation follows the decision rights at each node rather than selecting the most attractive number anywhere in the tree.
Interpret the value as a model-dependent bound
The University of Manchester notes on games develop the distinction between game models and their solutions. Here choosing Left ensures a score of at least three against either available MIN response. Against MIN’s best response, the score is exactly three. This does not say the realised score must always be three: MIN could choose seven. Nor does it mean Left maximises an average under an unstated probability distribution for MIN’s actions. Worst-case evaluation and probabilistic prediction are different objectives.
Understand what alpha-beta pruning preserves
After fully evaluating Left at three, suppose the search examines the Right leaf with value two first. MIN can already hold Right to at most two, so MAX will not prefer it to Left regardless of the other Right leaf. A correct alpha-beta procedure can avoid examining that remaining leaf without changing the minimax value. The benefit depends on search order and tree structure. Pruning is justified by bounds on relevance to the final comparison, not by guessing that an unexplored outcome is unlikely.
Separate exact search from heuristic cutoffs
If a larger tree cannot be searched to its terminal outcomes, an evaluation function may estimate values at a depth cutoff. Backing those estimates up computes a minimax value for the truncated approximation, not an exact solution of the complete game. An important consequence may lie just beyond the cutoff. Record the depth, evaluation rule and any approximation when reporting such a result. A precisely implemented algorithm can still provide an imperfect answer when its input values approximate unresolved future branches.
Distinguish information and equilibrium concepts
Perfect information concerns observation of earlier moves at decision points; it is not identical to knowing everyone’s preferences, often called complete information. Chance moves can also exist in a perfect-information game, but they require probability-weighted evaluation at chance nodes rather than treating chance as MIN. The MIT lecture on learning and minimax places the zero-sum theorem alongside broader equilibrium concepts. A game-tree algorithm and the general mixed-strategy minimax theorem are related ideas, but their representations and assumptions should not be collapsed into one unexplained claim.
Game-tree verification checklist
- Identify who controls each decision node.
- State whether information is perfect and chance is present.
- Define terminal utilities and opposing preferences.
- Apply minimum or maximum backups at the correct nodes.
- Distinguish worst-case values from predicted averages.
- Justify pruning with valid bounds.
- Disclose heuristic evaluations and search cutoffs.
For the four-leaf example, manually checking both MIN choices is enough to verify the root value. A larger implementation should preserve the same transparent semantics: every returned number should be traceable to terminal values or explicitly labelled estimates, with the player controlling each comparison clearly identified.
Frequently asked questions
Why does the largest terminal score not determine the minimax choice?
Because another player may control whether it is reached. The calculation respects each player’s available responses.
Does correct alpha-beta pruning change the minimax answer?
No. It omits branches that cannot affect the answer, assuming the pruning bounds and search are implemented correctly.
Does perfect information mean there is no chance?
No. Perfect information concerns observation of previous moves. A model can include observed chance outcomes and needs suitable probability calculations at chance nodes.
Sources and verification
- Game playing: minimax, alpha-beta pruning and evaluation — MIT course handout — 2026-10-23
- Games, Games, Games: notes for the game theory course — Andrea Schalk, University of Manchester — 2026-10-23
- Learning in games: Foundations, 24 September 2026 — MIT 6.7980 lecture notes — 2026-10-23