Game Playing Techniques Like Minimax Procedure

Definition

The minimax procedure is a decision-making algorithm used in adversarial games in which one player tries to maximize the score or utility, while the other tries to minimize it. It works by exploring the game tree, assigning values to terminal positions or estimated values to non-terminal positions, and then propagating those values upward to determine the best move for the current player.

In simple words, minimax assumes that:

the maximizing player always chooses the move with the highest possible benefit,
the minimizing player always chooses the move that gives the lowest possible benefit to the maximizing player.

This makes minimax especially suitable for competitive games where both sides are rational and goal-directed.

Main Content

1. Game Trees and State Space Representation

Game tree concept

A game can be represented as a tree where each node corresponds to a possible game state, and each edge corresponds to a legal move. The root node represents the current board or position, and each level alternates between moves by the two players. This structure is called a game tree.

State space and branching

The total set of all possible positions reachable from a given starting state is called the state space. The number of possible moves from a node is called the branching factor. In many games, the branching factor is very large, which makes complete exploration difficult. For example, chess has a huge branching factor, which is why practical AI systems cannot examine every possible continuation to the end of the game.

A small game tree for illustration:

                     A
                  /     \
                B         C
              /   \      /   \
             D     E    F     G

Here, A is the current position, B and C are possible moves, and D, E, F, G are later outcomes. In actual games, the tree may have millions of nodes.

2. Minimax Principle

Max and Min levels

In minimax, the algorithm alternates between two types of nodes:

MAX nodes, where the AI tries to maximize the utility value
MIN nodes, where the opponent tries to minimize the utility value

If the AI is the first player, it starts at a MAX node. After the AI’s move, the opponent moves from a MIN node, and the process continues.

Backing up values

At the terminal states, each outcome is assigned a numerical value, such as:

+1 for win
0 for draw
-1 for loss

These values are then propagated upward. At a MAX node, the highest child value is selected. At a MIN node, the lowest child value is selected. This backward propagation helps determine the best action from the current state.

Example:

                    MAX
                  /     \
                MIN     MIN
               /   \    /   \
              3     5   2     9

Left MIN chooses min(3, 5) = 3
Right MIN chooses min(2, 9) = 2
MAX chooses max(3, 2) = 3

So the best move for MAX leads to the left subtree.

3. Evaluation Functions and Practical Game Playing

Need for heuristic evaluation

In many real games, it is not possible to search until the final terminal state because the game tree is too large. Therefore, the search is stopped at a reasonable depth, and the resulting positions are evaluated using an evaluation function or heuristic function. This function estimates how good a position is for a player.

Examples of evaluation features

In chess, an evaluation function may consider:

material advantage (pieces remaining)
mobility (number of legal moves)
king safety
control of the center
pawn structure

In tic-tac-toe, evaluation might be simpler:

number of winning lines open
whether a player has two in a row
whether the board is a draw state

Because the evaluation function approximates the true value of a board state, the strength of a game-playing program often depends heavily on how well this function is designed.

Working / Process

1. Build the game tree from the current state

Start with the present game position as the root node and generate all legal moves for the current player. Each legal move creates child nodes, and the tree alternates between MAX and MIN levels.

2. Assign values to terminal or cut-off nodes

If a node is a final game result, give it a utility value such as win, loss, or draw. If the search stops before reaching the end, apply a heuristic evaluation function to estimate the position’s value.

3. Propagate values upward and choose the best move

Move back up the tree from the leaves to the root. At MAX nodes, choose the maximum child value. At MIN nodes, choose the minimum child value. The action associated with the best root value becomes the selected move.

A simple flow can be shown as:

Current State
      |
Generate Moves
      |
Build Game Tree
      |
Evaluate Leaf Nodes
      |
Back Up Min/Max Values
      |
Choose Best Move

Advantages / Applications

Optimal decision-making under perfect play assumptions

Minimax gives the best possible move when both players are assumed to play rationally and the game is deterministic with complete information.

Wide use in board and strategy games

It is used in chess engines, checkers programs, tic-tac-toe solvers, Othello agents, and many other adversarial AI systems. It forms the foundation for more advanced methods such as alpha-beta pruning.

Conceptually simple and mathematically clear

The method is easy to understand, easy to implement for small games, and provides a clear framework for reasoning about competitive decision problems in AI.

Summary

Minimax is a game-playing technique for choosing the best move in adversarial games.
It works by treating one player as MAX and the opponent as MIN.
The algorithm searches a game tree, evaluates outcomes, and selects the move with the best guaranteed value.

Important terms to remember

game tree, state space, MAX node, MIN node, utility function, evaluation function, heuristic, branching factor.