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Minimax decision
Definition
A minimax decision is an action selected because it leads to the best value the player can secure against optimal opposing play. In a conventional game tree, the maximizing player chooses an action whose successor state has the highest minimax value. The decision is the chosen move; the value is its numerical evaluation.
What it means
Several related ideas are easy to mix up. Minimax is the decision principle, minimax search is a procedure for evaluating continuations, the minimax value is the resulting score, and the minimax decision is the action returned for execution.
A decision therefore needs an action identity as well as a number. Knowing that the best attainable value is four does not tell a game agent which piece to move. An implementation must preserve the link between a root action and its evaluated continuation. When several actions attain the same best value, the minimax criterion alone does not specify a unique choice.
How it works
List the legal actions at the current state. For each action, evaluate the successor under the rules of the game, using minimum backups on opposing turns and maximum backups on the player's turns. Select an action attaining the largest successor value when the current player is MAX.
In notation, the decision belongs to argmax over actions a of V(Result(s,a)). The word argmax refers to the action attaining a maximum, whereas max refers to the maximum number. If the decision belongs to MIN instead, the direction reverses while the utility perspective remains fixed.
This classical interpretation assumes a deterministic, two-player, zero-sum, perfect-information setting. With heuristic cutoffs, the action is best according to the searched tree and evaluation function; it need not be the true full-game minimax decision. Tie-breaking, legality checks, and search completion status should be explicit.
Three practical examples
- Game controls: convert evaluated board positions into one legal move that the agent can actually play.
- Explanation interfaces: display the selected action alongside the strongest modeled opposing reply.
- Search testing: verify that an algorithm returns the correct action when several branches have similar or tied scores.
Think of it this way
Think of choosing a package after checking the least favorable item an opponent could force you to receive from each package. The chosen package is the decision; the quality of its least favorable item is the supporting value.
The analogy does not imply that every decision consists of only two turns. A game may contain many future choices, and the evaluation must account for who controls each of them.
Worked example
MAX can choose North, East, or West. Each action gives MIN a choice between two terminal outcomes. North has utilities 7 and 2, East has 4 and 4, and West has 9 and 4. These utilities are all measured for MAX.
The successor minimax values are respectively 2, 4, and 4. The maximum value is 4, and the set of minimax decisions is {East, West}. Returning the number 4 alone would not identify an executable move. Choosing East by a documented alphabetical or fixed-order tie rule is a valid minimax decision; choosing West is also valid.
Suppose the system uses a secondary preference for the larger favorable outcome when worst-case values tie. It would choose West because 9 exceeds 4. That secondary rule does not change the minimax value and should not be misreported as something minimax uniquely requires.
Now revise West's second terminal utility from 4 to 3 while keeping everything else unchanged. West's worst-case value becomes 3, so East is the unique minimax decision. Choosing West because its best leaf remains 9 would violate the stated criterion. This small sensitivity check helps distinguish robust action selection from optimistic leaf selection.
Illustrative case study
Fictional case: a developer builds a training-game assistant that correctly computes a root score but occasionally executes a different move. Investigation shows that the interface stores the highest score without retaining the associated action identifier.
The team changes the result record to include the selected move, score, search depth, and whether the result is exact or heuristic. It also documents how ties are resolved. The fictional case concerns the connection between reasoning and action; no measured reliability gain is claimed.
Common misconceptions and mistakes
A minimax decision is not necessarily unique. Different actions can offer identical protection while differing in features outside the main utility model. A tie rule can select among them, but the additional preference should be disclosed.
Do not confuse an action with a complete strategy. A root decision specifies what to do now. A strategy specifies choices across future contingencies. Recomputing after each move can provide subsequent decisions, but one stored root move is not a full contingent plan.
Nor should a timeout result be labeled exact merely because its score is numeric. If some relevant alternatives remain unexplored and no valid bounds settle the choice, the returned action may only be provisional.
Business use
The distinction matters in game products, decision-support interfaces, and adversarial simulations where a recommendation must become an executable step. Record the action, its evaluated value, the assumptions, and the scope of completed search.
Test action selection separately from score calculation. Include tied actions, all-negative utilities, illegal moves, and trees where the most attractive individual leaf lies behind a poor opposing response. These checks expose errors that a single reported root score can hide.
People also ask
Can two different moves both be minimax decisions?
Yes. If both attain the best minimax value, either satisfies the primary criterion.
Is argmax the same as max?
No. Argmax identifies an action or argument; max gives its numerical value.
Does a minimax decision promise a favorable outcome?
No. The best available worst-case result may still be a loss or a negative utility.
Check your understanding
Root actions A, B, and C have minimax values −6, −2, and −4. What should MAX choose?
Answer: B. The largest value is −2 even though all three values are negative.
Related terms
Learn next
implement or trace a tiny game tree while keeping action identifiers beside scores. Then compare a single selected move with a strategy specifying responses to every reachable opposing choice.
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Sources
Poole and Mackworth, Artificial Intelligence: Foundations of Computational Agents, discusses game-tree action selection and adversarial search: https://artint.info/3e/html/ArtInt3e.Ch14.S3.html
Their game-representation chapter distinguishes a run from a contingent strategy: https://artint.info/3e/html/ArtInt3e.Ch14.S2.html
The tied-action example and fictional implementation case are original teaching illustrations.
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