adversarial search
TRANSCRIPT
ARTIFICIAL INTELLIGENCE
Chapter 6Adversarial search
Prepared by:Issam Al-dehedhawyFarah Al-Tufaili
ADVERSARIAL SEARCH
Examining the problems that arise when we try to plan ahead in a world where other agents are planning against us.
OVERVIEW OF GAMES
Games are a form of multi-agent environment
What do other agents do and how do they affect our success?
Cooperative vs. competitive multi-agent environments .
Competitive multi-agent environments give rise to adversarial search.
OUTLINES Introduction to game
Optimal decisions in games
Optimal strategies
Tic-tac-toe.
The minimax algorithm.
Optimal decisions in multiplayer games.
GAME SEARCHGame-playing programs developed by AI researchers since the beginning of the modern AI
– Programs playing chess, checkers, etc (1950s)
• Specifics of the game search:
– Sequences of player’s decisions we control– Decisions of other player(s) we do not control
• Contingency problem: – Many possible opponent’s moves must be “covered” by the solution– Opponent’s behavior introduces an uncertainty in the game– We do not know exactly what the response is going to be
• Rational opponent – maximizes it own utility (payoff) Function.
GAMES VS. SEARCHSearch – no adversary -Solution is (heuristic) method for finding goal -Heuristics and CSP techniques can find optimal solution Evaluation function: estimate of cost from start to goal through given node.
Examples: path planning, scheduling activities
Games – adversary -Solution is strategy (strategy specifies move for every possible opponent reply).
-Time limits force an approximate solution to be taken.Evaluation function: evaluate “goodness” of game position
Examples: chess, checkers, Othello, backgammon
TYPES OF GAME PROBLEMSTypes of game problems
Adversarial games:
win of one player is a loss of the other
– Cooperative games:
players have common interests and utility function
– A spectrum of game problems in between the two
TYPES OF GAME PROBLEMS Fully cooperative games
Adversarial games
we focus on adversarial games only!!
OPTIMAL DECISIONS IN GAMES• Game problem formulation:
– Initial state: initial board position and identifies the player to move
– successor function: legal moves a player can make
– Goal (terminal test): determines when the game is over
– Utility (payoff) function: measures the outcome of the
game and its desirability
• Search objective:
– find the sequence of player’s decisions (moves)
maximizing its utility (payoff)
– Consider the opponent’s moves and their utility
OPTIMAL STRATEGIES
-Find the contingent strategy for MAX assuming an infallible MIN opponent.
-Assumption: Both players play optimally !!
Given a game tree, the optimal strategy can be determined by using the minimax value of each node:
-MINIMAX-VALUE(n)=
UTILITY(n) If n is a terminal
maxs successors(n) MINIMAX-VALUE(s) If n is a max then
mins successors(n) MINIMAX-VALUE(s) If n is a min node
EXAMPLE OF AN ADVERSARIAL 2 PERSON GAME:TIC-TAC-TOEPlayer 1 (x) moves
first
win
drow
loss
A two player game where minmax algorithm is applies is Tic–Tac-Toe. In this game in order to win you must fill a row,a column or diagonal
(X or O).
GAME PROBLEM FORMULATION (TIC-TAC-TOE)
Objectives: •Player 1:
maximize outcome
•Player 2:minimize outcome
-1 0 1
Terminal (goal) states
Utility:
Minimax is a decision rule algorithm, which is represented as a game-tree.
It has applications in decision theory, game theory , statistics and philosophy.
Minimax is applied in two player games. The one is the min and the other is the max player.
By agreement the root of the game-tree represents the max player.
It is assumed that each player aims to do the best move for himself and therefore the worst move for his opponent in order to win the game.
MINIMAX ALGORITHM
MINIMAX ALGORITHM
How to deal with the contingency problem?
• Assuming that the opponent is rational and always optimizes its behavior (opposite to us) we consider the best response. opponent’s
• Then the minimax algorithm determines the best move
MINIMAX ALGORITHM
3 8 12 2 4 6 14 5 2
3 2 2
3Max
Min
Utility
MINIMAX ALGORITHM
4 3 6 2 1 9 5 3 1
Max
Min
Utility 2 7 5
Max
MINIMAX ALGORITHM
2 1 9 5 3 1
Max
Min
Utility 2 7 5
Max
4
4 3 6
MINIMAX ALGORITHM
2 1 9 5 3 1
Max
Min
Utility 2 7 5
Max
4 6
4 3 6
MINIMAX ALGORITHM
4 3 6 2 1 9 5 3 1
Max
Min
Utility 2 7 5
Max
4
64
MINIMAX ALGORITHM
4 3 6 2 1 9 5 3 1
Max
Min
2 7 5
Max
4
64 2
Utility
MINIMAX ALGORITHM
4 3 6 2 1 9 5 3 1
Max
Min
2 7 5
Max
4
64 2 9
Utility
MINIMAX ALGORITHM
4 3 6 2 1 9 5 3 1
Max
Min
2 7 5
Max
4
64 2 9
2
Utility
MINIMAX ALGORITHM
2 1 9 5 3
3
Max
Min
2 7 5
Max
4
64 2 9
2
1Utili
ty 4 3 6
MINIMAX ALGORITHM
2 1 9 5 3
3
Max
Min
2 7 5
Max
4
64 2 9
2
1
7
Utility 4 3 6
MINIMAX ALGORITHM
2 1 9 5 3
3
Max
Min
2 7 5
Max
4
64 2 9
2
1
7
3
Utility 4 3 6
MINIMAX ALGORITHM
2 1 9 5 3
3
Max
Min
2 7 5
Max
4
64 2 9
2
1
7
3
4
Utility 4 3 6
MINIMAX ALGORITHMfunction MINIMAX-DECISION(state) returns an action
inputs: state, current state in game vMAX-VALUE(state)
return the action in SUCCESSORS(state) with value v
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)) return 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)) return v
PROPERTIES OF MINIMAX
Complete? Yes (if tree is finite)
Optimal? Yes (against an optimal opponent)
Time complexity? O(bm)
Space complexity? O(bm) (depth-first exploration,for algorithm that generates all successors at once or O(m) for an algorithm that generates suuccesors one at atime )
For chess, b ≈ 35, m ≈100 for "reasonable" games exact solution completely infeasible
Minimax advantages:
-Returns an optimal action, assuming perfect opponent play.
Minimax is the simplest possible (reasonable) game search algorithm.
-Tic-Tac-Toe player, chances are you either looked this up on Wikipedia, or
invented it in the process.)
Minimax disadvantages: It's completely infeasible in practice.
! When the search tree is too large, we need to limit the search depth and
apply an evaluation function to the cut-off states.
MULTIPLAYER GAMES
Games allow more than two players
Single minimax values become vectors