Introduction to Evaluation Function of Minimax Algorithm in Game Theory

Last Updated : 20 Apr, 2023

Prerequisite: Minimax Algorithm in Game Theory
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, Backgammon, 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 a Heuristic Function.
The evaluation function is unique for every type of game. In this post, the 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 the maximizer turn or a low value for the board if the minimizer turn.
For this scenario let us consider X as the maximizer and O as the minimizer.
Let us build our evaluation function :

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

evaluation_function1

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

evaluation_function2

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

evaluation_function3

We could have chosen any positive/negative. For the sake of simplicity, we chose 10 and shall use lowercase ‘x’ and lowercase ‘o’ to represent the players and an underscore ‘_’ to represent a blank space on the board.
If we represent our board as a 3x3 2D character matrix, like char board[3][3]; then we have to check each row, each column, and the diagonals to check if either of the players has gotten 3 in a row.

C++

// C++ program to compute evaluation function for // Tic Tac Toe Game. #include<stdio.h> #include<algorithm> using namespace std;  // Returns a value based on who is winning // b[3][3] is the Tic-Tac-Toe board int evaluate(char b[3][3]) {     // Checking for Rows for X or O victory.     for (int row = 0; row<3; row++)     {         if (b[row][0]==b[row][1] && b[row][1]==b[row][2])         {             if (b[row][0]=='x')                return +10;             else if (b[row][0]=='o')                return -10;         }     }      // Checking for Columns for X or O victory.     for (int col = 0; col<3; col++)     {         if (b[0][col]==b[1][col] && b[1][col]==b[2][col])         {             if (b[0][col]=='x')                 return +10;             else if (b[0][col]=='o')                 return -10;         }     }      // Checking for Diagonals for X or O victory.     if (b[0][0]==b[1][1] && b[1][1]==b[2][2])     {         if (b[0][0]=='x')             return +10;         else if (b[0][0]=='o')             return -10;     }     if (b[0][2]==b[1][1] && b[1][1]==b[2][0])     {         if (b[0][2]=='x')             return +10;         else if (b[0][2]=='o')             return -10;     }      // Else if none of them have won then return 0     return 0; }  // Driver code int main() {     char board[3][3] =     {         { 'x', '_', 'o'},         { '_', 'x', 'o'},         { '_', '_', 'x'}     };      int value = evaluate(board);     printf("The value of this board is %d\n", value);     return 0; }

Java

// Java program to compute evaluation function for // Tic Tac Toe Game.  class GFG {  // Returns a value based on who is winning // b[3][3] is the Tic-Tac-Toe board static int evaluate(char b[][]) {     // Checking for Rows for X or O victory.     for (int row = 0; row < 3; row++)     {         if (b[row][0] == b[row][1] && b[row][1] == b[row][2])         {             if (b[row][0] == 'x')             return +10;             else if (b[row][0] == 'o')             return -10;         }     }      // Checking for Columns for X or O victory.     for (int col = 0; col < 3; col++)     {         if (b[0][col] == b[1][col] && b[1][col] == b[2][col])         {             if (b[0][col] == 'x')                 return +10;             else if (b[0][col] == 'o')                 return -10;         }     }      // Checking for Diagonals for X or O victory.     if (b[0][0] == b[1][1] && b[1][1] == b[2][2])     {         if (b[0][0] == 'x')             return +10;         else if (b[0][0] == 'o')             return -10;     }     if (b[0][2] == b[1][1] && b[1][1] == b[2][0])     {         if (b[0][2] == 'x')             return +10;         else if (b[0][2] == 'o')             return -10;     }      // Else if none of them have won then return 0     return 0; }  // Driver code public static void main(String[] args) {     char board[][] =     {         { 'x', '_', 'o'},         { '_', 'x', 'o'},         { '_', '_', 'x'}     };      int value = evaluate(board);     System.out.printf("The value of this board is %d\n", value); } }  // This code is contributed by PrinciRaj1992

Python3

# Python3 program to compute evaluation  # function for Tic Tac Toe Game.   # Returns a value based on who is winning  # b[3][3] is the Tic-Tac-Toe board  def evaluate(b):        # Checking for Rows for X or O victory.      for row in range(0, 3):                if b[row][0] == b[row][1] and b[row][1] == b[row][2]:                        if b[row][0] == 'x':                 return 10              else if b[row][0] == 'o':                  return -10       # Checking for Columns for X or O victory.      for col in range(0, 3):                if b[0][col] == b[1][col] and b[1][col] == b[2][col]:                        if b[0][col]=='x':                 return 10              else if b[0][col] == 'o':                  return -10       # Checking for Diagonals for X or O victory.      if b[0][0] == b[1][1] and b[1][1] == b[2][2]:                if b[0][0] == 'x':              return 10          else if b[0][0] == 'o':              return -10            if b[0][2] == b[1][1] and b[1][1] == b[2][0]:                if b[0][2] == 'x':              return 10          else if b[0][2] == 'o':              return -10            # Else if none of them have won then return 0      return 0    # Driver code  if __name__ == "__main__":        board = [['x', '_', 'o'],               ['_', 'x', 'o'],               ['_', '_', 'x']]            value = evaluate(board)      print("The value of this board is", value)   # This code is contributed by Rituraj Jain

// C# program to compute evaluation function for // Tic Tac Toe Game. using System;  class GFG {  // Returns a value based on who is winning // b[3,3] is the Tic-Tac-Toe board static int evaluate(char [,]b) {     // Checking for Rows for X or O victory.     for (int row = 0; row < 3; row++)     {         if (b[row, 0] == b[row, 1] && b[row, 1] == b[row, 2])         {             if (b[row, 0] == 'x')             return +10;             else if (b[row, 0] == 'o')             return -10;         }     }      // Checking for Columns for X or O victory.     for (int col = 0; col < 3; col++)     {         if (b[0, col] == b[1, col] && b[1, col] == b[2, col])         {             if (b[0, col] == 'x')                 return +10;             else if (b[0, col] == 'o')                 return -10;         }     }      // Checking for Diagonals for X or O victory.     if (b[0, 0] == b[1, 1] && b[1, 1] == b[2, 2])     {         if (b[0, 0] == 'x')             return +10;         else if (b[0, 0] == 'o')             return -10;     }     if (b[0, 2] == b[1, 1] && b[1, 1] == b[2, 0])     {         if (b[0, 2] == 'x')             return +10;         else if (b[0, 2] == 'o')             return -10;     }      // Else if none of them have won then return 0     return 0; }  // Driver code public static void Main(String[] args) {     char [,]board =     {         { 'x', '_', 'o'},         { '_', 'x', 'o'},         { '_', '_', 'x'}     };      int value = evaluate(board);     Console.Write("The value of this board is {0}\n", value); } }  // This code is contributed by Rajput-Ji

JavaScript

<script> // Javascript program to compute evaluation function for // Tic Tac Toe Game.  // Returns a value based on who is winning // b[3][3] is the Tic-Tac-Toe board function evaluate(b) {      // Checking for Rows for X or O victory.     for (let row = 0; row < 3; row++)     {         if (b[row][0] == b[row][1] && b[row][1] == b[row][2])         {             if (b[row][0] == 'x')                 return +10;             else if (b[row][0] == 'o')                 return -10;         }     }        // Checking for Columns for X or O victory.     for (let col = 0; col < 3; col++)     {         if (b[0][col] == b[1][col] && b[1][col] == b[2][col])         {             if (b[0][col] == 'x')                 return +10;             else if (b[0][col] == 'o')                 return -10;         }     }        // Checking for Diagonals for X or O victory.     if (b[0][0] == b[1][1] && b[1][1] == b[2][2])     {         if (b[0][0] == 'x')             return +10;         else if (b[0][0] == 'o')             return -10;     }     if (b[0][2] == b[1][1] && b[1][1] == b[2][0])     {         if (b[0][2] == 'x')             return +10;         else if (b[0][2] == 'o')             return -10;     }        // Else if none of them have won then return 0     return 0; }  // Driver code let board=[[ 'x', '_', 'o'],         [ '_', 'x', 'o'],         [ '_', '_', 'x']];  let value = evaluate(board); document.write("The value of this board is "+ value+"<br>");  // This code is contributed by avanitrachhadiya2155 </script>

Output

The value of this board is 10

Time Complexity: O(max(row,col))
Auxiliary Space: O(1)

The idea of this article is to understand how to write a simple evaluation function for the game Tic-Tac-Toe. In the next article we shall see how to combine this evaluation function with the minimax function. Stay Tuned.
This article is written by Akshay L. Aradhya.

Minimax Algorithm in Game Theory | Set 1 (Introduction)

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Introduction to Evaluation Function of Minimax Algorithm in Game Theory

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