int alphaBeta( int alpha, int beta, int depthleft ) {
   if( depthleft == 0 ) return quiesce( alpha, beta );
   for ( all moves)  {
      score = -alphaBeta( -beta, -alpha, depthleft - 1 );
      if( score >= beta )
         return beta;   //  fail hard beta-cutoff
      if( score > alpha )
         alpha = score; // alpha acts like max in MiniMax
   }
   return alpha;
}

score = alphaBetaMax(-oo, +oo, depth);

int alphaBetaMax( int alpha, int beta, int depthleft ) {
   if ( depthleft == 0 ) return evaluate();
   for ( all moves) {
      score = alphaBetaMin( alpha, beta, depthleft - 1 );
      if( score >= beta )
         return beta;   // fail hard beta-cutoff
      if( score > alpha )
         alpha = score; // alpha acts like max in MiniMax
   }
   return alpha;
}
 
int alphaBetaMin( int alpha, int beta, int depthleft ) {
   if ( depthleft == 0 ) return -evaluate();
   for ( all moves) {
      score = alphaBetaMax( alpha, beta, depthleft - 1 );
      if( score <= alpha )
         return alpha; // fail hard alpha-cutoff
      if( score < beta )
         beta = score; // beta acts like min in MiniMax
   }
   return beta;
}

Minimax is an algorithm used to determine the score in a zero-sum game after a certain number of moves, with best play according to an evaluation function.

int maxi( int depth ) {
    if ( depth == 0 ) return evaluate();
    int max = -oo;
    for ( all moves) {
        score = mini( depth - 1 );
        if( score > max )
            max = score;
    }
    return max;
}
 
int mini( int depth ) {
    if ( depth == 0 ) return -evaluate();
    int min = +oo;
    for ( all moves) {
        score = maxi( depth - 1 );
        if( score < min )
            min = score;
    }
    return min;
}

Negamax is a common way of implementing minimax and derived algorithms.

• Instead of using two separate subroutines for the Min player and the Max player, it passes on the negated score due to following mathematical relation:

max(a, b) == -min(-a, -b)

This is how the pseudo-code of the recursive algorithm looks like. For clarity move making and unmaking is omitted.

int negaMax( int depth ) {
    if ( depth == 0 ) return evaluate();
    int max = -oo;
    for ( all moves)  {
        score = -negaMax( depth - 1 );
        if( score > max )
            max = score;
    }
    return max;
}

http://web.archive.org/web/20120421170110/http://chessprogramming.wikispaces.com/Alpha-Beta

http://web.archive.org/web/20120209041607/http://chessprogramming.wikispaces.com/Minimax

http://web.archive.org/web/20120209042116/http://chessprogramming.wikispaces.com/Negamax