Games provide well-defined test beds for developing AI techniques. They have clear rules and allow direct comparison of algorithms against humans. Game playing involves searching game trees to find optimal strategies, taking into account opponents' possible moves. The minimax algorithm and its improved version alpha-beta pruning are commonly used to search game trees and select the best next move.

1. Game Playing in Artificial Intelligence

2. 1. Games are fun
2. They have well-defined rules
3. They provide advanced, existing test-beds for developing
several ideas.
4. It’s a good reasoning problem, formal and nontrivial.
5. Direct comparison with humans and other computer
programs is easy
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3. 1. Sequence of moves to play
2. Rules that specify possible moves
3. Rules that specify a payment for each move
4. Objective is to maximize your payment
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4. 4
 Unpredictable opponent  specifying a move
for every possible opponent reply
 Time limits  unlikely to find goal, must
approximate

5. 5
Opponent’s Move
Generate New Position
Generate Successors
Game
Over?
Evaluate Successors
Move to Highest-Valued Successor
Game
Over?
no
no yes
yes
Two-Player Game

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7.  The term "game" means a sort of conflict in which n individuals or groups
(known as players) participate.
 A list of "rules" stipulates the conditions under which the game begins.
 A game is said to have "perfect information" if all moves are known to each of
the players involved.
 A "strategy" is a list of the optimal choices for each player at every stage of a
given game.
 A "move" is the way in which game progresses from one stage to
another, beginning with an initial state of the game to the final state. The total
number of moves constitute the completeness of the game.
 The payoff or outcome, refers to what happens at the end of a game.
 Minimax - good outcomes.
 Maximin - bad outcomes.
 The primary game theory is the Mini-Max Theorem says:
"If a Minimax of one player corresponds to a Maximin of the other
player, then that outcome is the best both players can hope for."

8.  Two players: A and B
 A moves first and they take turns until the game is over.
Winner gets award, loser gets penalty.
 Games as search:
◦ Initial state: e.g. board configuration of chess
◦ Successor function: list of (move, state) pairs specifying
legal moves.
◦ Terminal test: Is the game finished?
◦ Utility function: Gives numerical value of terminal states.
E.g. win (+1), lose (-1) and draw (0)
 A uses search tree to determine next move.
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 MiniMax rule: Decision Making in Multi-agent Systems
 Perfect play for deterministic, perfect-information games.
• Idea: make the move for player MAX which has the most
benefit assuming that MIN makes the best move for MIN
in response
• This is computed by a recursive process
• The backed-up value of each node in the tree is
determined by the values of its children
• For a MAX node, the backed-up value is the
maximum of the values of its children (i.e. the best
for MAX)
• For a MIN node, the backed-up value is the minimum
of the values of its children (i.e. the best for MIN)

10. 10
The computer is Max.
The opponent is Min.
At the leaf nodes, the
utility function
is employed. Big value
means good, small is bad.
computer’s
turn
opponent’s
turn
computer’s
turn
opponent’s
turn
leaf nodes
are evaluated

11. 11
 utility function: the function applied to leaf
nodes
 backed-up value
◦ of a max-position: the value of its largest successor
◦ of a min-position: the value of its smallest successor
 minimax procedure: search down several
levels; at the bottom level apply the utility
function, back-up values all the way up to the
root node, and that node selects the move.

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Consider a 2-ply (two step) game:
Max want’s largest outcome --- Min want’s smallest.
1. Start with the current position as a MAX node.
2. Expand the game tree a fixed number of ply (half-moves).
3. Apply the evaluation function to the leaf positions.
4. Calculate back-up up values bottom-up.
5. Pick the move which was chosen to give the MAX value at the root.

13. 13

14.  Complete? Yes (if tree is finite)
 Optimal? Yes (against an optimal opponent)
 Time complexity? O(bm)
 Space complexity? O(bm) (depth-first exploration)
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15.  Eliminating a branch without consideration is called
pruning
◦ Want to visit as many board states as possible (Can be used for entire
search or cutoff search)
 Want to avoid whole branches (prune them)
 Because they can’t possibly lead to a good score
 A way to improve the performance of the Minimax Procedure
 Basic idea: “If you have an idea which is surely bad, don’t take the
time to see how truly awful it is”
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2 7 1
=2
>=2
<=1
?
• We don’t need to compute the
value at this node.

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 3MAX
MIN =3
3 12 8 2
X X
 2
14
 14
5
 5
2
=2
=3

17.  Traverse the search tree in depth-first order
 For each MAX node n, α(n)=maximum child value found
so far
◦ Starts with –
◦ Increases if a child returns a value greater than the current
α(n)
◦ Lower-bound on the final value
 For each MIN node n, β(n)=minimum child value found
so far
◦ Starts with +
◦ Decreases if a child returns a value less than the current
β(n)
◦ Upper-bound on the final value
 MAX cutoff rule: At a MAX node n, cut off search if
α(n)>=β(n)
 MIN cutoff rule: At a MIN node n, cut off search if
β(n)<=α(n)
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function ALPHA-BETA-SEARCH(state) returns an action
inputs: state, current state in game
vMAX-VALUE(state, - ∞ , +∞)
return the action in SUCCESSORS(state) with value v
function MAX-VALUE(state, , ) returns a utility value
if TERMINAL-TEST(state) then return UTILITY(state)
v  - ∞
for a,s in SUCCESSORS(state) do
v  MAX(v,MIN-VALUE(s,  , ))
if v ≥  then return v
  MAX( ,v)
return v

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function MIN-VALUE(state,  , ) returns a utility value
if TERMINAL-TEST(state) then return UTILITY(state)
v  + ∞
for a,s in SUCCESSORS(state) do
v  MIN(v,MAX-VALUE(s,  , ))
if v ≤  then return v
  MIN( ,v)
return v

20.  Properties
 Pruning does not affect final
result

 Good move ordering improves
effectiveness of pruning

 With "perfect ordering," time
complexity = O(bm/2)
 doubles depth of search
 A simple example of the value
of reasoning about which
computations are relevant (a
form of metareasoning)

 Effectiveness
 Guaranteed to compute same
root value as Minimax
 Worst case: no pruning, same
as Minimax (O(bd))
 Best case: when each player’s
best move is the first option
examined, you examine only
O(bd/2) nodes, allowing you
to search twice as deep!
 For Deep Blue, alpha-beta
pruning reduced the average
branching factor from 35-40
to 6.
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APPLICATIONS
 Training Simulators
 Military simulations
 Management simulations
 Economic simulations
 Education
 Entertainment
 Virtual Environments
Movies