Implement Tic-Tac-Toe or any multiplayer game using min-max algorithm

Experiment No: AI-

TITLE: - Implement Tic-Tac-Toe or any multiplayer game  minmax algorithm

Problem Statement: - Implement Tic-Tac-Toe multiplayer game using AI technique & MinMax algorithm (C++/Java)

Objective:

To study techniques of Artificial Intelligence & MinMax algorithm

Requirements (Hw/Sw): PC, Netbeans IDE/Text Editor (Linux)

 Theory:-

Since the invention of computers or machines, their capability to perform various tasks went on growing exponentially. Humans have developed the power of computer systems in terms of their diverse working domains, their increasing speed, and reducing size with respect to time.

A branch of Computer Science named Artificial Intelligence pursues creating the computers or machines as intelligent as human beings.

1956->John McCarthy coined the term Artificial Intelligence. Demonstration of the first running AI program at Carnegie Mellon University.

What is Artificial Intelligence?

According to the father of Artificial Intelligence, John McCarthy, it is “The science and engineering of making intelligent machines, especially intelligent computer programs”.

Artificial Intelligence is a way of making a computer, a computer-controlled robot, or a software think intelligently, in the similar manner the intelligent humans think.

AI is accomplished by studying how human brain thinks, and how humans learn, decide, and work while trying to solve a problem, and then using the outcomes of this study as a basis of developing intelligent software and systems.

Philosophy of AI

While exploiting the power of the computer systems, the curiosity of human, lead him to wonder, “Can a machine think and behave like humans do?”

Thus, the development of AI started with the intention of creating similar intelligence in machines that we find and regard high in humans.

Goals of AI

·        To Create Expert Systems − The systems which exhibit intelligent behavior, learn, demonstrate, explain, and advice its users.

·        To Implement Human Intelligence in Machines − Creating systems that understand, think, learn, and behave like humans.

·                   Programming Without and With AI

·         The programming without and with AI is different in following ways −

Programming Without AI	Programming With AI
A computer program without AI can answer the specific questions it is meant to solve.	A computer program with AI can answer the generic questions it is meant to solve.
Modification in the program leads to change in its structure.	AI programs can absorb new modifications by putting highly independent pieces of information together. Hence you can modify even a minute piece of information of program without affecting its structure.
Modification is not quick and easy. It may lead to affecting the program adversely.	Quick and Easy program modification.

Implementation of Tic-Tac-Toe game

Rules of the Game

§  The game is to be played between two people (in this program between HUMAN and COMPUTER).

§  One of the player chooses ‘O’ and the other ‘X’ to mark their respective cells.

§  The game starts with one of the players and the game ends when one of the players has one whole row/ column/ diagonal filled with his/her respective character (‘O’ or ‘X’).

§  If no one wins, then the game is said to be draw.

§  Implementation
In our program the moves taken by the computer and the human are chosen randomly. We use rand() function for this.

Winning Strategy

If both the players play optimally then it is destined that you will never lose (“although the match can still be drawn”). It doesn’t matter whether you play first or second.In another ways – “ Two expert players will always draw ”.

Minimax Algorithm in Game Theory

Minimax is a kind of backtracking algorithm that is used in decision making and game theory to find the optimal move for a player, assuming that your opponent also plays optimally. It is widely used in two player turn based games such as Tic-Tac-Toe, Backgamon, Mancala, Chess, etc.

In Minimax the two players are called maximizer and minimizer. The maximizer tries to get the highest score possible while the minimizer tries to get the lowest score possible while minimizer tries to do opposite.

Every board state has a value associated with it. In a given state if the maximizer has upper hand then, the score of the board will tend to be some positive value. If the minimizer has the upper hand in that board state then it will tend to be some negative value. The values of the board are calculated by some heuristics which are unique for every type of game.

Example:
Consider a game which has 4 final states and paths to reach final state are from root to 4 leaves of a perfect binary tree as shown below. Assume you are the maximizing player and you get the first chance to move, i.e., you are at root, and your opponent at next level. Which move you would make as a maximizing player considering that your opponent also plays optimally?

Since this is a backtracking based algorithm, it tries all possible moves, then backtracks and makes a decision.

§  Maximizer goes LEFT : It is now the minimizers turn. The minimizer now has a choice between 3 and 5. Being the minimizer it will definitely choose the least among both, that is 3

§  Maximizer goes RIGHT : It is now the minimizers turn. The minimizer now has a choice between 2 and 9. He will choose 2 as it is the least among the two values.

Being the maximizer you would choose the larger value that is 3. Hence the optimal move for the maximizer is to go LEFT and the optimal value is 3.

Now the game tree looks like below :

The above tree shows two possible scores when maximizer makes left and right moves.

Evaluation Function

As seen in the above article, each leaf node had a value associated with it. We had stored this value in an array. But in the real world when we are creating a program to play Tic-Tac-Toe, Chess, Backgamon, etc. we need to implement a function that calculates the value of the board depending on the placement of pieces on the board. This function is often known as Evaluation Function. It is sometimes also called Heuristic Function.

The evaluation function is unique for every type of game. In this post, evaluation function for the game Tic-Tac-Toe is discussed. The basic idea behind the evaluation function is to give a high value for a board if maximizer‘s turn or a low value for the board if minimizer‘s turn.

For this scenario let us consider X as the maximizer and O as the minimizer.

Let us build our evaluation function :

1.     If X wins on the board we give it a positive value of +10.

2.     If O wins on the board we give it a negative value of -10.

3.     If no one has won or the game results in a draw then we give a value of +0.

We could have chosen any positive / negative value other than 10. For the sake of simplicity we chose 10 for the sake of simplicity we shall use lower case ‘x’ and lower case ‘o’ to represent the players and an underscore ‘_’ to represent a blank space on the board.

Minimax Algorithm in Game Theory | (Tic-Tac-Toe AI – Finding optimal move)

Let us combine what we have learnt so far about minimax and evaluation function to write a proper Tic-Tac-Toe AI (Artificial Intelligence) that plays a perfect game. This AI will consider all possible scenarios and makes the most optimal move.

Finding the Best Move :

We shall be introducing a new function called findBestMove(). This function evaluates all the available moves using minimax() and then returns the best move the maximizer can make. The pseudocode is as follows :

function findBestMove(board):

    bestMove = NULL

    for each move in board :

        if current move is better than bestMove

            bestMove = current move

    return bestMove

Minimax :

To check whether or not the current move is better than the best move we take the help of minimax() function which will consider all the possible ways the game can go and returns the best value for that move, assuming the opponent also plays optimally
The code for the maximizer and minimizer in the minimax() function is similar to findBestMove() , the only difference is, instead of returning a move, it will return a value. Here is the pseudocode :

function minimax(board, depth, isMaximizingPlayer):

    if current board state is a terminal state :

        return value of the board

    if isMaximizingPlayer :

        bestVal = -INFINITY 

        for each move in board :

            value = minimax(board, depth+1, false)

            bestVal = max( bestVal, value) 

        return bestVal

    else :

        bestVal = +INFINITY 

        for each move in board :

            value = minimax(board, depth+1, true)

            bestVal = min( bestVal, value) 

        return bestVal

Checking for GameOver state :

To check whether the game is over and to make sure there are no moves left we use isMovesLeft() function. It is a simple straightforward function which checks whether a move is available or not and returns true or false respectively. Pseudocode is as follows :

function isMovesLeft(board):

    for each cell in board:

        if current cell is empty:

            return true

    return false

Making our AI smarter :

One final step is to make our AI a little bit smarter. Even though the following AI plays perfectly, it might choose to make a move which will result in a slower victory or a faster loss. Lets take an example and explain it.

Assume that there are 2 possible ways for X to win the game from a give board state.

§  Move A : X can win in 2 move

§  Move B : X can win in 4 moves

§   

Our evaluation function will return a value of +10 for both moves A and B. Even though the move A is better because it ensures a faster victory, our AI may choose B sometimes. To overcome this problem we subtract the depth value from the evaluated score. This means that in case of a victory it will choose a the victory which takes least number of moves and in case of a loss it will try to prolong the game and play as many moves as possible. So the new evaluated value will be

§  Move A will have a value of +10 – 2 = 8

§  Move B will have a value of +10 – 4 = 6

§   

Now since move A has a higher score compared to move B our AI will choose move A over move B. The same thing must be applied to the minimizer. Instead of subtracting the depth we add the depth value as the minimizer always tries to get, as negative a value as possible. We can subtract the depth either inside the evaluation function or outside it. Anywhere is fine. I have chosen to do it outside the function. Pseudocode implementation is as follows.

if maximizer has won:

    return WIN_SCORE – depth

else if minimizer has won:

    return LOOSE_SCORE + depth 

Outcome:

Output :

The value of the best Move is : 10

The Optimal Move is :

ROW: 2 COL: 2

This image depicts all the possible paths that the game can take from the root board state. It is often called the Game Tree.
The 3 possible scenarios in the above example are :

• Left Move: If X plays [2,0]. Then O will play [2,1] and win the game. The value of this move is -10
• Middle Move: If X plays [2,1]. Then O will play [2,2] which draws the game. The value of this move is 0
• Right Move: If X plays [2,2]. Then he will win the game. The value of this move is +10;

Remember, even though X has a possibility of winning if he plays the middle move, O will never let that happen and will choose to draw instead.

Therefore the best choice for X, is to play [2,2], which will guarantee a victory for him.

1. Click Here to Download TicTacToe Java program

2. Click Here to Download TicTacToe Java program

Conclusion: