GAME PLAYING

The search problems we have discussed so far have been 
deterministic -- the next state is completely determined 
by the current state and the action taken.

In games this is frequently not true -- either there is an 
opponent or random element which affects state.

Playing Against an Opponent

Finding the perfect solution

Minimax algorithm:

	Generate the whole game tree.

	Apply a utility function to each leaf node.

	Working from bottom up determine the expected utility 
	for each node in the tree:

		For a branch in the tree where it is the opponents 
		move, assume the opponent will make the move 
		that is best for them -- attach the minimum utility 
		value of child nodes.

		For a branch in the tree where it is our (the 
		system's) move, select the best move -- attach 
		the maximum utility value of the child nodes.

	When get to top of tree, the best move (assuming an 
	opponent that does not make mistakes) and its utility 
	have been computed.



			IMPERFECT DECISIONS

We cannot always (or even very often) compute the entire 
search tree in order to identify the optimal move 
against an opponent.

In order to determine what appears to be the best move 
without completely computing solution we need an 
evaluation function to compare states.

The evaluation function must:

	agree with utility function for terminal states

	not take too long

	accurately reflect the chances of winning.

In chess, material advantage can be used as an evaluation 
function.



		DECIDING WHEN TO CUT OFF SEARCH

quiescence search -- identify states likely to have large 
swings in quality in future moves and search more 
from them.

horizon problem -- some unavoidable action can always be 
pushed just beyond the depth of the search.


Alpha-Beta Pruning

Idea: can infer that more search is useless in some 
situations.

When a better solution is guaranteed elsewhere in tree, 
there is no need to evaluate more along current path.



			DEALING WITH CHANCE

Use probability of random actions within search tree.

If rolling a 6-sided die, there is a 1/6 chance of each path at 
a die-roll branch.

Can combine chance and opponents for games such as 
backgammon.


			SEARCH IN LISP


(defun tree-search (states goal-p successors combiner)
	(cond
		((null states) `())
		((funcall goal-p (first states)) (first states))
		(`t (tree-search
			(funcall combiner
				(funcall successors (first states))
				(rest states))
			goal-p successors combiner))))

(defun DFS (start goal-p successors)
	(tree-search (list start) goal-p successors #'append))

(defun in-Houston? (state) (equal state `Houston))

(defparam *moves* 
	`((CS -> Hempstead)
	  (CS -> Madisonville)
	  (Madisonville -> Houston)
	  (Madisonville -> Dallas)
	  (Hempstead -> Houston)
	  (Hempstead -> Austin))

(defun next-cities (city)
	(let ((stops `()))
		(dolist (element *moves* stops)
			(if (equal city (first element))
				 (setf stops (append (last element) stops))))))

(DFS `College-Station #'in-Houston? #'next-cities)