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Adversarial Search


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Outline

Adversarial Search

Game Theory

Formal Definition of Game Optimal decisions

- pruning

Imperfect, real-time decisions


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ADVERSARIAL SEARCH

Competitive environments,in which

the agents goals are in conflict,give riseto adversarial search problems often

known as games.


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GamesMathematical Game Theory,a branch ofeconomics,views any multiagent environment as a

game

Agents are cooperative or competitive.

AI games are,two-player, turn-taking and fullyobservable environments

utility values at the end of the game are alwaysequal and opposite. For example,if one player winsthe game of chess(+1),the other player necessarilyloses(-1). It is this opposition between the agents

utility functions that makes the situation adversarial.


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Formal Definition of GameConsider the games (Tic-tac-toe) with two players,whom we will callMAX and MIN. MAX moves first,and then they take turns moving until

the game is over. At the end of the game, points are awarded to thewinning player and penalties are given to the loser. A game can beformally defined as a search problem with the following components :

o The initial state,which includes the board position and

identifies the player to move.o A successor function,which returns a list of (move,state)pairs,each indicating a legal move and the resulting state.o A terminal test,which describes when the game is over.States where the game has ended are called terminal

states.oA utility function (also called an objective function orpayoff function),which give a numeric value for the terminalstates. In chess,the outcome is a win,loss,or draw,with

values +1,-1,or 0.


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Games vs. search problems

"Unpredictable" opponent specifying amove for every possible opponent reply

Time limits unlikely to find goal, mustapproximate


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Game tree (2-player,deterministic, turns)


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Minimax

Perfect play for deterministic games

Idea: choose move to position with highest minimax

value= best achievable payoff against best play

E.g., 2-ply game:


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function minimax(node,depth)if depth


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Minimax algorithm


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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 chess, b 35, m 100 for "reasonable" games exact solution completely infeasible


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- pruning example


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- pruning example


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- pruning example


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- pruning example


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- pruning example


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Properties of -

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 whichcomputations are relevant (a form of metareasoning)


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Why is it called -?

is the value of thebest (i.e., highest-value) choice foundso far at any choicepoint along the pathfor max

If vis worse than ,maxwill avoid it

prune that branch


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The - algorithm


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The - algorithm


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Resource limits

Suppose we have 100 secs, explore 104nodes/sec106nodes per move

Standard approach:

cutoff test:

e.g., depth limit (perhaps add quiescence search)

evaluation function


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Evaluation functions

For chess, typically linear weighted sum of features

Eval(s) = w1 f1(s) + w2 f2(s) + + wn fn(s)

e.g., w1 = 9 with

f1(s) = (number of white queens) (number of blackqueens), etc.


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Cutting off search

MinimaxCutoffis identical to MinimaxValueexcept1. Terminal?is replaced by Cutoff?2. Utilityis replaced by Eval3.

Does it work in practice?

bm = 106, b=35 m=4

4-ply lookahead is a hopeless chess player!

4- l human novice


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Deterministic games in practice

Checkers: Chinook ended 40-year-reign of human world championMarion Tinsley in 1994. Used a precomputed endgame databasedefining perfect play for all positions involving 8 or fewer pieces onthe board, a total of 444 billion positions.

Chess: Deep Blue defeated human world champion Garry Kasparovin a six-game match in 1997. Deep Blue searches 200 millionpositions per second, uses very sophisticated evaluation, andundisclosed methods for extending some lines of search up to 40ply.

Othello: human champions refuse to compete against computers,who are too good.

Go: human champions refuse to compete against computers, whoare too bad. In go, b > 300, so most programs use patternknowledge bases to suggest plausible moves.


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Summary

Games are fun to work on!

They illustrate several important pointsabout AI

perfection is unattainable mustapproximate

good idea to think about what to think

about

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