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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

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
- 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

- 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

- Tic-tac-toe
- Connect Four
- Checkers

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

- 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

- 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]

- for example, in most chess books each piece is given a value

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 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

- 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

- 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

- 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

- 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]

- can stop expanding a sub-tree when a value less than the best-so-far is found
- 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]

- can stop expanding a sub-tree when a value greater than best-so-far is found

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

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

- 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

- 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)

- avoids the horizon effect

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

- 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

- 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

- scaling up by dumbing down

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