Minimax on a Simple 3-Level Two-Player Game Tree: Step-by-Step Selection of the Maximizer’s Best Move

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Nov 3, 2026

In a deterministic, perfect-information, two-player game, Minimax computes the best move by assuming:

the maximizing player will choose moves that maximize the final outcome,
the minimizing player will choose moves that minimize it.

We’ll build a simple game tree with 3 plies (3 levels of decisions) and then show exactly how Minimax “backs up” leaf utilities to pick the best move for the maximizing player.

Key ideas are captured by:

Game tree
Terminal utility
Max node
Min node
Minimax

We interpret utilities as “good for the maximizing player.” So higher numbers are better for Max.

Important: “3 levels” can be interpreted in multiple ways (edges/plies vs. node-depth). Here we use three decision levels: Max (root) → Min → Max, and the utilities appear at the final leaves.

Pro Tip: Name node types before computing

Before evaluating anything, label each internal node as Max node or Min node. Minimax becomes a mechanical “max at Max nodes, min at Min nodes” backup.

Minimax value recursion (backup rule)

Let $V(n)$ be the minimax value of node $n$ .

If $n$ is a Max node: $V(n) = \max_{c \in \text{children}(n)} V(c)$
If $n$ is a Min node: $V(n) = \min_{c \in \text{children}(n)} V(c)$

At the leaves, $V(\text{leaf})$ is the given Terminal utility.

Compute Minimax values and select Max’s best move

1
Step 1
Use the given leaf utilities at the bottom of the tree: under A1: {3,5}, under A2: {2,9}, under B1: {0,4}, under B2: {6,1}.
2
Step 2
At A1 (Max): $V(A1)=\max(3,5)=5$ . At A2 (Max): $V(A2)=\max(2,9)=9$ . At B1 (Max): $V(B1)=\max(0,4)=4$ . At B2 (Max): $V(B2)=\max(6,1)=6$ .
3
Step 3
At A (Min): $V(A)=\min(V(A1),V(A2))=\min(5,9)=5$ . At B (Min): $V(B)=\min(V(B1),V(B2))=\min(4,6)=4$ .
4
Step 4
At Root (Max): $V(R)=\max(V(A),V(B))=\max(5,4)=5$ .
5
Step 5
Since $V(A)=5$ and $V(B)=4$ , Max chooses move A at the root.

Minimax backup results (leaf → internal nodes → root)

Values computed by max on Max nodes and min on Min nodes.

What the numbers mean (intuitively)

Under move A, the minimizing player will choose the worse option for Max between A1 and A2. Since A1 leads to 5 and A2 leads to 9 (from Max’s perspective), Min picks 5, making $V(A)=5$ .
Under move B, Min compares B1 (4) vs. B2 (6), and picks the smaller, 4, so $V(B)=4$ .
Therefore Max prefers A because $5 > 4$ .

Common pitfall: mixing up max and min at levels

A frequent mistake is applying max at Min nodes (or vice versa). In this course section: Root is Max, level 2 is Min, level 3 is Max; only then do you take max/min in that pattern.

Quick checks and extensions

From game-tree construction to decision

Define terminal utilities

Assign numeric outcomes at leaf states (higher is better for the maximizing player)."

Backup at deepest Max nodes

Compute each Max node as the maximum of its leaf children."

Backup at Min nodes

Compute each Min node as the minimum of its already-evaluated children."

Choose at the root Max node

Pick the move with the larger backed-up value at the root."

Knowledge Check

Question 1 of 4

Q1Single choice

In Minimax, which operation is used at a Max node?

Take the minimum of child values

Take the maximum of child values

Always choose the leftmost child

Average the child values

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