I’ve recently tried to understand how minimax can be optimised. I found a great code but I was asking myself: is this code working for a tree with only three different steps ?
I’m not sure I’ve understood precisely how alpha and beta are being updated.

     Max
   /  |  
  Min Min Min
 /|   |   | 
9 2 4  1   4 1 2

def max_value(board, alpha, beta):
    """
    Returns the maximum value for the current player on the board 
    using alpha-beta pruning.
    """
    if terminal(board):
        return utility(board)
    v = -math.inf
    for action in actions(board):
        v = max(v, min_value(result(board, action), alpha, beta))
        alpha = max(alpha, v)
        if alpha >= beta:
            break
    return v

def min_value(board, alpha, beta):
    """
    Returns the minimum value for the current player on the board 
    using alpha-beta pruning.
    """
    if terminal(board):
        return utility(board)
    v = math.inf
    for action in actions(board):
        v = min(v, max_value(result(board, action), alpha, beta))
        beta = min(beta, v)
        if alpha >= beta:
            break
    return v

def minimax(board):
    """
    Returns the optimal action for the current player on the board 
    using the minimax algorithm with alpha-beta pruning.
    """
    if terminal(board):
        return None

    if player(board) == X:
        v = -math.inf
        opt_action = None
        for action in actions(board):
            new_value = min_value(result(board, action), -math.inf, math.inf)
            if new_value > v:
                v = new_value
                opt_action = action
        return opt_action
    
    elif player(board) == O:
        v = math.inf
        opt_action = None
        for action in actions(board):
            new_value = max_value(result(board, action), -math.inf, math.inf)
            if new_value < v:
                v = new_value
                opt_action = action
        return opt_action