utility functions minimax search

Hi I'm confused how you can determine the utility functions on with a minimax search Explain it with any game that you can use a minimax search with Basically i am asking开发者_JS百科 how do you determine the utility functions Cheers

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The utility value is just some arbitrary value that the player receives when arriving at a certain state in the game. For instance, in Tic-tac-toe, your utility function could simply be 1 for a win, 0 for a tie, or -1 for a loss. Running minmax on this would at best find a set of actions that result in 1 (a win).

Another example would be chess (not that you can feasibly run minimax on a game of chess). Say your utility function comes from a certain number that is based on the value of the piece you captured or lost

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Determining the utility value of a move at a certain state has to do with the experience of the programmer and his/her knowledge of the game.
Utility values on a terminal state are kind of easy to determine. In Tic-tac-toe, for instance, a terminal state for player X is when the Xs are aligned in diagonal, vertically, or horizontally. Any move that creates such a state is a terminal state and you can create a function that checks that. If it is a terminal state, the function returns a 1 or -1. If your player agent is player X and after player X's move it determines that player O will win, then the function returns a -1. The function returns a 1 if it determines that it is its own winning move. If all cells are occupied with the last possible move and nobody has won, then the function returns a zero.

This is at terminal states only. It is critical to evaluate intermediate states because, even in a 3x3 game, there are lots of combinations to consider. If you include symmetrical moves you have 9! possible states in Tic-tac-toe. For those intermediate cases, you need to come up with an evaluation function that returns a score for each state as they related to other states.

Suppose that I assign the terminal state values of 810, 0, and -810. For each move, the score would be 810 / (# of moves). So if I reach a terminal state in 6 moves, the score would be 810/6 = 135. In 9 moves, the score would be 90. An evaluation function fashioned this way would favor moves that reach a terminal state faster. However, it still evaluates to a leaf node. We need to evaluate before reaching a leaf node, though, but this could also be part of an evaluation function. Supposed that, in the game below, player 1 is X. So X moves next. The following are the legal moves (row, column) for X: (1) 0,0 (2) 0,2 (3) 2,0 (4) 2,1 (5) 2,2

| |O| | |O|X|X| | | | |

The utility value for each move should favor the best moves. The best moves, in this case, are either (2) or (5). So an evaluation function will assign a utility value of 81, for instance to each of those. Move (4) is the worst possible move for the X player (and would also warranty that you lose the game against an intelligent player) so the function would assign a value of -9 to that move. Moves (1) and (3), while not ideal, will not make you lose, so we might assign a 1.
So when minimax evaluates those 5 moves, because your player X, is max, the choice would be either (2) or (5).

If we focus on options (2) or (5), the game will be on a terminal state two moves after these. So, in reality, the evaluation function should look 2 moves ahead of the current legal moves to return the utility values. (This strategy follows the lines of depth limited search, where your function evaluates at a certain depth and produces a utility value without reaching a leaf node - or terminal state)

Now I'll circle back to my first statement. The utility value will be determined by an evaluation function coded per the programmer's knowledge of the game.

Hopefully, I'm not confusing you...