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Game Playing. Chapter 5. Game playing. Search applied to a problem against an adversary some actions are not under the control of the problem-solver there is an opponent (hostile agent) Since it is a search problem, we must specify states & operations/actions

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game playing

Game Playing

Chapter 5

game playing2
Game playing
  • Search applied to a problem against an adversary
    • some actions are not under the control of the problem-solver
    • there is an opponent (hostile agent)
  • Since it is a search problem, we must specify states & operations/actions
    • initial state = current board; operators = legal moves; goal state = game over; utility function = value for the outcome of the game
    • usually, (board) games have well-defined rules & the entire state is accessible
basic idea
Basic idea
  • Consider all possible moves for yourself
  • Consider all possible moves for your opponent
  • Continue this process until a point is reached where we know the outcome of the game
  • From this point, propagate the best move back
    • choose best move for yourself at every turn
    • assume your opponent will make the optimal move on their turn
examples
Examples
  • Tic-tac-toe
  • Connect Four
  • Checkers
problem
Problem
  • For interesting games, it is simply not computationally possible to look at all possible moves
    • in chess, there are on average 35 choices per turn
    • on average, there are about 50 moves per player
    • thus, the number of possibilities to consider is 35100
solution
Solution
  • Given that we can only look ahead k number of moves and that we can’t see all the way to the end of the game, we need a heuristic function that substitutes for looking to the end of the game
    • this is usually called a static board evaluator (SBE)
    • aperfect static board evaluator would tell us for what moves we could win, lose or draw
    • possible for tic-tac-toe, but not for chess
creating a sbe approximation
Creating a SBE approximation
  • Typically, made up of rules of thumb
    • for example, in most chess books each piece is given a value
      • pawn = 1; rook = 5; queen = 9; etc.
    • further, there are other important characteristics of a position
      • e.g., center control
    • we put all of these factors into one function, weighting each aspect differently potentially, to determine the value of a position
      • board_value =  * material_balance +  * center_control + … [the coefficients might change as the game goes on]
compromise
Compromise
  • If we could search to the end of the game, then choosing a move would be relatively easy
    • just use minimax
  • Or, if we had a perfect scoring function (SBE), we wouldn’t have to do any search (just choose best move from current state -- one step look ahead)
  • Since neither is feasible for interesting games, we combine the two ideas
basic idea9
Basic idea
  • Build the game tree as deep as possible given the time constraints
  • apply an approximate SBE to the leaves
  • propagate scores back up to the root & use this information to choose a move
  • example
score percolation minimax
Score percolation: MINIMAX
  • When it is my turn, I will choose the move that maximizes the (approximate) SBE score
  • When it is my opponent’s turn, they will choose the move that minimizes the SBE
    • because we are dealing with competitive games, what is good for me is bad for my opponent & what is bad for me is good for my opponent
    • assume the opponent plays optimally [worst-case assumption]
minimax algorithm
MINIMAX algorithm
  • Start at the the leaves of the trees and apply the SBE
  • If it is my turn, choose the maximum SBE score for each sub-tree
  • If it is my opponent’s turn, choose the minimum score for each sub-tree
  • The scores on the leaves are how good the board appears from that point
  • Example
alpha beta pruning
Alpha-beta pruning
  • While minimax is an effective algorithm, it can be inefficient
    • one reason for this is that it does unnecessary work
    • it evaluates sub-trees where the value of the sub-tree is irrelevant
    • alpha-beta pruning gets the same answer as minimax but it eliminates some useless work
    • example
      • simply think: would the result matter if this node’s score were +infinity or -infinity?
cases of alpha beta pruning
Cases of alpha-beta pruning
  • Min level (alpha-cutoff)
    • can stop expanding a sub-tree when a value less than the best-so-far is found
      • this is because you’ll want to take the better scoring route [example]
  • Max level (beta-cutoff)
    • can stop expanding a sub-tree when a value greater than best-so-far is found
      • this is because the opponent will force you to take the lower-scoring route [example]
alpha beta algorithm
Alpha-beta algorithm
  • Maximizer’s moves have an alpha value
    • it is the current lower bound on the node’s score (i.e., max can do at least this well)
    • if alpha >= beta of parent, then stop since opponent won’t allow us to take this route
  • Minimizer’s moves have a beta value
    • it is the current upper bound on the node’s score (i.e., it will do no worse than this)
    • if beta <= alpha of parent, then stop since we (max) will won’t choose this
slide17
Use
  • We project ahead k moves, but we only do one (the best) move then
  • After our opponent moves, we project ahead k moves so we are possibly repeating some work
  • However, since most of the work is at the leaves anyway, the amount of work we redo isn’t significant (think of iterative deepening)
alpha beta performance
Alpha-beta performance
  • Best-case: can search to twice the depth during a fixed amount of time [O(bd/2) v. O(bd)]
  • Worst-case: no savings
    • alpha-beta pruning & minimax always return the same answer
    • the difference is the amount of work they do
    • effectiveness depends on the order in which successors are examined
      • want to examine the best first
  • Graph of savings
refinements
Refinements
  • Waiting for quiescence
    • avoids the horizon effect
      • disaster is lurking just beyond our search depth
      • on the nth move (the maximum depth I can see) I take your rook, but on the (n+1)th move (a depth to which I don’t look) you checkmate me
    • solution
      • when predicted values are changing frequently, search deeper in that part of the tree (quiescence search)
secondary search
Secondary search
  • Find the best move by looking to depth d
  • Look k steps beyond this best move to see if it still looks good
  • No? Look further at second best move, etc.
    • in general, do a deeper search at parts of the tree that look “interesting”
  • Picture
book moves
Book moves
  • Build a database of opening moves, end games, tough examples, etc.
  • If the current state is in the database, use the knowledge in the database to determine the quality of a state
  • If it’s not in the database, just do alpha-beta pruning
ai games
AI & games
  • Initially felt to be great AI testbed
  • It turned out, however, that brute-force search is better than a lot of knowledge engineering
    • scaling up by dumbing down
      • perhaps then intelligence doesn’t have to be human-like
    • more high-speed hardware issues than AI issues
    • however, still good test-beds for learning
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