The MinMax algorithm in AI, popularly known as the minimax, is a backtracking algorithm used in decision making, game theory and artificial intelligence (AI). It is used to find the optimal move for a player, assuming that the opponent is also playing optimally. Popular two-player computer or online games like Chess, Tic-Tac-Toe, Checkers, Go, etc. use this algorithm.
A backtracking algorithm is used to find a solution to computational problems in such a way that a candidate is incrementally built towards a solution, one step at a time. And the candidate that fails to complete a solution is immediately abandoned.
How does it work?
In the MinMax algorithm in artificial intelligence , there are two players, Maximiser and Minimiser. Both these players play the game as one tries to get the highest score possible or the maximum benefit while the opponent tries to get the lowest score or the minimum benefit.
Every game board has an evaluation score assigned to it, so the Maximiser will select the maximised value, and the Minimiser will select the minimised value with counter moves. If the Maximiser has the upper hand, then the board score will be a positive value, and if the Minimiser has the upper hand, then the board score will be a negative value.
This is based on the zero-sum game concept where the total utility score gets divided between the two players. Thus, an increase in one player’s score leads to a decrease in the opponent player’s score, making the total score always zero. So, for one player to win, the other has to lose.
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Breaking down the MinMax algorithm in AI
The complete game tree is explored with a depth-first search algorithm in the MinMax algorithm in AI. It proceeds entirely down to the terminal node of the tree and then backtracks through the tree.
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The goal is to find the best possible move for a player. This can be done by choosing the node with the best evaluation score. The best choice will be made after evaluating all the potential moves of the opponent. The algorithm looks ahead at all the possible values till the end and makes a decision for the player.
The game tree above is a nested data structure that is used to evaluate the moves. Here the root node is Level 0, which branches out into Level 1 or parent nodes, which further branch out into Level 2 or child nodes. The branching out can continue to many levels, having the potential of infinite levels. Level 0 is like the current state of the board, while Level 1 is all the possible states of boards depending on the next move.
Thus, if Player 2 has made a move, we can assume that the root node is the current state of the board, waiting for Player 1’s move. Level 1 nodes contain all the possible moves for Player 1, and the Level 2 nodes contain all the possible moves for Player 2 based on each possible move of Player 1.
Consider an example where there are four final states and the path to reach these are from the root to the four leaves of a tree. The values of the four leaves are 3, 6 on the left and 4, 7 on the right. It is the Maximiser/Player 1’s turn to make a move. To run through the algorithm, assumptions for each move have to be made.
If the Player 1 chooses to go left, the Minimiser/Player 2 has to choose the least between 3 and 6, and so they would choose 3. Whereas if the Player 1 chooses right, the Player 2 will choose 4, which is the minimum of the two values, 4 and 7. So, Level 1 now has the values 3 and 4.
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Since it is the Player 1/Maximiser’s turn, they have to choose the maximum of Level 1 nodes. Thus, they will choose 3. Then the optimal choice is to go left.
The steps for the MinMax algorithm in artificial intelligence can be stated as follows:
- Create the entire game tree.
- Evaluate the scores for the leaf nodes based on the evaluation function.
- Backtrack from the leaf to the root nodes:
For Maximizer, choose the node with the maximum score.
For Minimizer, choose the node with the minimum score.
- At the root node, choose the node with the maximum value and select the respective move.
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Pseudo-code for MinMax Algorithm
function minMax(node, depth, maximizingPlayer)
if depth = 0 or node is a terminal node
return the heuristic value of node
if maximizingPlayer
bestValue := -∞
for each child of node
value := minMax(child, depth – 1, FALSE)
bestValue := max(bestValue, value)
return bestValue
else
bestValue := +∞
for each child of node
value := minMax(child, depth – 1, TRUE)
bestValue := min(bestValue, value)
return bestValue
In this pseudo-code:
- ‘node’ represents the current state of the game or decision tree node.
- ‘depth’ represents the maximum depth to which the algorithm should search.
- ‘maximizingPlayer’ is a Boolean variable indicating whether the current player is maximizing or minimizing.
- The ‘if depth = 0 or node is a terminal node’ condition checks if the maximum depth is reached or if the current node is a terminal node (i.e., end of the game).
- The ‘maximizingPlayer’ condition determines whether the algorithm should maximize or minimize the value.
- The algorithm recursively explores the game tree by evaluating all possible moves up to a certain depth and computes the heuristic value of each node.
- The ‘max’ and ‘min’ functions are used to select the maximum or minimum value among child nodes depending on whether the player is maximizing or minimizing.
Properties of MinMax algorithm in AI
Here I have listed the properties of MinMax algorithm in AI:
- The algorithm is complete, meaning in a finite search tree, a solution will be certainly found.
- It is optimal if both the players are playing optimally.
- Due to Depth-First Search (DFS) for the game tree, the time complexity of the algorithm is O(bm), where b is the branching factor and m is the maximum depth of the tree.
- Like DFS, the space complexity of this algorithm is O(bm).
Advantages
- A thorough assessment of the search space is performed.
- Decision making in AI is easily possible.
- New and smart machines are developed with this algorithm.
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Limitations
- Because of the huge branching factor, the process of reaching the goal is slower.
- Evaluation and search of all possible nodes and branches degrades the performance and efficiency of the engine.
- Both the players have too many choices to decide from.
- If there is a restriction of time and space, it is not possible to explore the entire tree.
But with Alpha-Beta Pruning, the algorithm can be improved.
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Applications of the MinMax Algorithm
The MinMax algorithm in AI is a fundamental component of game theory and artificial intelligence that is used in many domains were making decisions in the face of uncertainty or competition is common. Taking a close look at a few important applications:
- Robotics and Path Planning
MinMax is used in robotics for problems related to path planning and navigation, particularly when robots must avoid obstacles or plan paths across changing situations. It aids robots in making decisions that consider potential roadblocks and uncertainty to accomplish their goals.
- Strategic Planning
In many different fields, such as commercial planning, cybersecurity, and military strategy, MinMax is utilized in strategic planning and decision-making processes. Making the best decisions in competitive contexts requires the analysis of various strategies and the prediction of opponents’ movements.
- Game Playing
When playing turn-based video games, the MinMax algorithm in AI is often used to calculate the best move for the player. This tactic is used in games where players attempt to outwit their opponent and improve their chances of winning, such as Connect Four, Othello, Chequers, and Chess.
- Healthcare
When considering treatment alternatives, MinMax algorithm might be useful in weighing the potential benefits and drawbacks of each option. It helps healthcare providers decide what’s best for their patients’ treatment.
Variations of the MinMax Algorithm
Several modifications and improvements have been made to the MinMax algorithm in artificial intelligence to address various issues or boost its effectiveness in particular situations. Here are a few noteworthy MinMax algorithm in artificial intelligence variations:
- Alpha-Beta Pruning
Probably the most well-known application of the MinMax method is alpha-beta pruning. It does this by removing branches from the MinMax search tree that aren’t likely to have an impact on the outcome. The algorithm reduces the search space and increases speed by pruning branches that are guaranteed to be worse than previously evaluated branches by keeping two additional values, alpha and beta.
- Negamax Algorithm
A single function that combines the maximizing and minimizing of players is how the Negamax algorithm streamlines the MinMax method. It takes advantage of the fact that in two-player zero-sum games, the opponent’s value is negated, and the current player’s position is worth the same. The algorithm becomes simpler to use and has less complexity because of this simplification.
- Iterative Deepening
Iterative Deepening is a technique to improve the efficiency of the MinMax search that progressively increases the search depth until a time limit, or a terminal node is reached. Because of this, the technique can find reasonably good moves quickly, which is useful for scenarios requiring quick decisions or for real-time applications.
MinMax Algorithm Future Developments
Game theory and artificial intelligence have long relied heavily on the MinMax algorithm. The algorithm may be enhanced in a few places, though. The following are a few likely future advances for MinMax algorithms:
- Deep Learning Integration
Deep learning combined with MinMax could lead to more advanced bots that can comprehend complex patterns and moves in games. Deep learning has demonstrated notable success in a variety of fields, including gaming (e.g., AlphaGo).
- Metaheuristic Optimization
MinMax algorithms could be enhanced by incorporating metaheuristic optimization techniques, such as genetic algorithms or simulated annealing, to explore the search space more effectively and find optimal or near-optimal solutions in complex decision-making problems.
- Hybrid Algorithms
Future developments may involve the creation of hybrid algorithms that combine the strengths of MinMax with other techniques, such as Monte Carlo Tree Search (MCTS) or reinforcement learning. When dealing with complicated decision-making circumstances, these hybrid techniques may provide superior scalability and performance.
Conclusion
This article explains all the aspects of the min-max algorithm in AI. First, an introduction of the theory is provided with examples of where it is used, after which there is a description of how the algorithm works in a game.
The algorithm is broken down to explain how a decision to make an optimal move is taken based on moves and counter moves of the players. The properties of the algorithm are then listed. Lastly, the advantages and disadvantages of the algorithm are provided.
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