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====== Chess - Programming - Minimax Algorithm ======

**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.

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

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.

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

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[[Chess:Programming:Minimax Algorithm:A program to illustrate the Minimax Algorithm|A program to illustrate the Minimax Algorithm]]

[[Chess:Programming:Minimax Algorithm:A Tic-Tac-Toe example program to illustrate the Minimax Algorithm|A Tic-Tac-Toe example program to illustrate the Minimax Algorithm]]

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===== Pseudocode =====

<code cpp>
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
</code>

----

===== 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 the root and your opponent at next level.
  * Which move you would make as a maximizing player considering that your opponent also plays optimally?

{{:chess:programming:minmax.png?400|}}

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:

{{:chess:programming:minmax1.png?400|}}

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

<WRAP info>
**NOTE:**  Even though there is a value of 9 on the right subtree, the minimizer will never pick that.

  * We must always assume that our opponent plays optimally.
</WRAP>