Foundations of Artificial Intelligence

CSE 573 -- Fall 2001


Game PlayingKautz 

Game Playing - Why?

1. Game playing is one of the oldest areas of AI. In 1950's Claude Shannon the inventor of information theory) and Alan Turing wrote some of the first chess programs. Since then steady progress has been made in many games and several programs can challenge world champions (two things led to this - speed of computers and better search methods).
2. Provides a structured task in which it is very easy to measure success or failure. True. What's nice about it is that typically games are well-defined.
3. Not much knowledge required intially ...thought that straightforward search would be the solution. Turned out not to be the case except for the simplest of games -- the search space is just too big.
4. Why is tic-tac-toe boring? Search space is small enough that most adults can determine the perfect move.
5. Focus has been on games of perfect information: tic-tac-toe, chess, checkers, Go, Othello, backgammon. Knowledge available to each person is the same. Games of imperfect information include poker (can't see the other's hand) and bridge.

Game Playing An AI Favorite

• structured task
• not initially thought to require large amounts of knowledge
• focus on games of perfect information

Game Playing

Initial State

Operators

Terminal Test

Utility Function

A game can be defined as a search problem with the following components:

We've already converted some games into search problems, e.g., 8-puzzle...

Initial State

initial board/position and an indication of whose move it is.

Operators

define the set of legal moves from any position

Terminal Test

determines when the game is over, hopefully the game is in a goal state (i.e., you have won).

Utility Function

gives a numeric outcome for the game. Example in tic-tac-toe and chess the outcome is win (+1), draw (0), lose(-1) Some games have wider payoffs. For example, backgammon has a payoff range from +192 to -192.

Game Playing

Initial State

is the initial board/position

Operators

define the set of legal moves from any position

Terminal Test

determines when the game is over

Utility Function

gives a numeric outcome for the game

Game Playing as Search

1#1

Winston, p. 102. Generic game tree. Original board; new boards, etc.

1. Want to find path from initial state to goal. What if we try to apply one of our usual search algorithms. Exhaustive search: definitely not practical. Heuristic search? Fine for single-player games. Search for sequence of moves that leads to winning position.

2. Two player games are different. Second player has some say about his/her moves and probably won't choose the move that helps the first player.

3. This kind of tree is referred to as an and-or tree. (regular search is an OR tree.) Want to find a subtree (starting at the root), such that exactly one branch is selected when it's Max'x move and every branch is considered at Min's move. Strategy has to anticipate anything that Min does. OR-for Max, can choose move 1 or 2 or 3. AND-at Min level; have to consider all of Min's possible moves.

Partial Search Tree for Tic-Tac-Toe

2#2

3#3

Simple Minimax

4#4

5#5

Simplified Minimax Algorithm

1. Expand the entire tree below the root.
2. Evaluate the terminal nodes as wins for the minimizer or maximizer.
3. Select an unlabeled node, 6#6, all of whose children have been assigned values. If there is no such node, we're done -- return the value assigned to the root.
4. If 6#6 is a minimizer move, assign it a value that is the minimum of the values of its children. If 6#6 is a maximizer move, assign it a value that is the maximum of the values of its children. Return to Step 3.

Another Example

7#7

1. In game tree search, a move is a pair of actions. One player's action is a ply. 2-ply = one move.
2. Called a minimax decision because it maximizes the utility under the assumption that the opponent will play perfectly to minimize it.
3. Time complexity: 8#8 (9#9 plies and 10#10 branching.) Impractical for e.g. chess (11#11 to 12#12).

The Need for Imperfect Decisions

Problem:

Minimax assumes the program has time to search to the terminal nodes.

Solution:

The Need for Imperfect Decisions

Problem:

Minimax assumes the program has time to search to the terminal nodes.

Solution:

Cut off search earlier and apply a heuristic evaluation function to the leaves.

13#13

Static Evaluation Functions Minimax depends on the translation of board quality into a single, summarizing number. Can be difficult. Expensive.

• Add up values of pieces each player has (weighted by importance of piece). E.g. intro chess: pawn = 1pnt; knight or bishop = 3pnts; rook = 5 pnts; and queen = 9pnts.
• Isolated pawns are bad. How well protected is your king? How much maneuverability to you have? Do you control the center of the board?
• How many plies with perfect eval function?

14#14

Design Issues of Heuristic Minimax

Evaluation Function:

What features should we evaluate and how should we use them? An evaluation function should:

15#15

Linear Evaluation Functions

• 16#16
• 17#17 -- weight; 18#18 -- feature
• Steps in designing an evaluation function:
  1. Pick informative features
  2. Tune the weights
  Deep Blue: precision in eval (normalized, beteen 0 -- 1) is 19#19 to 20#20 ! (lots of fine-tuning is important)

21#21

Creating Evaluation Functions

• Features / weights for chess?
• How tune weights?
• How find features?

22#22

Design Issues of Heuristic Minimax

Search:

search to a constant depth

Problems:

23#23

Design Issues of Heuristic Minimax

• Some portions of the game tree may be ``hotter'' than others. Should search to quiescence. Continue along a path as long as one move's static value stands out (indicating a likely capture).
• Horizon effect
• Secondary search. (singular extension heuristic)