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Turn-Based Games. sources: http://www.game-research.com/ www.gamespot.com Wikipedia.org Russell & Norvig AI Book; Chapter 5 (and slides) My own. H é ctor Mu ñ oz-Avila. Turn-Based Strategy Games. Early strategy games was dominated by turn-based games Derivate from board games Chess

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Turn based games l.jpg

Turn-Based Games

  • sources:

  • http://www.game-research.com/

  • www.gamespot.com

  • Wikipedia.org

  • Russell & Norvig AI Book; Chapter 5 (and slides)

  • My own

Héctor Muñoz-Avila


Turn based strategy games l.jpg
Turn-Based Strategy Games

  • Early strategy games was dominated by turn-based games

    • Derivate from board games

      • Chess

      • The Battle for Normandy (1982)

      • Nato Division Commanders (1985)

  • Turn-based strategy:

  • game flow is partitioned in turns or rounds.

  • Turns separate analysis by the player from actions

  • “harvest, build, destroy” in turns

  • Two classes:

    • Simultaneous

    • Mini-turns


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Turn-Based Games Continues to be A Popular Game Genre

  • At least 3 sub-styles are very popular:

    • “Civilization”-style games

      • Civilization IV came out last week

    • Fantasy-style (RPG)

      • Heroes of Might and Magic series

    • Poker games

      • Poker Academy


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Some Historical Highlights

  • 1952 Turing design a chess algorithm. Around the same time Claude Shannon also develop a chess program

  • 1956 Maniac versus Human

  • 1970 Hamurabi. A game about building an economy for a kingdom

  • The Battle for Normandy (1982)

  • 1987 Pirates!

  • 1990 Civilization

  • 1995 HoMM

  • 1996 Civilization II

    • The best game ever?

  • 2005 Civilization IV

  • 2006 HoMM V


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Side-tracking: Game Design: Contradicting Principles

  • Principle: All actions can be done from a single screen.

  • Classical example: Civ IV

  • But: HoMM uses two interfaces: HoMM IV


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Coming back: How to Construct Good AI?

  • Idea: Lets just use A* and define a good heuristic for the game

    • Search space: a bipartite tree

    • After all didn’t we use it with the 9-puzzle game?

  • Problems with this idea:

  • Adversarial: we need to consider possible moves of our opponent (s)

  • Time limit: (think Chess)


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Types of AdversarialTBGs (from AI perspective)

Chance

Deterministic

Chess, Go, rock-paper-scissors

Perfect

information

Backgammon, monopoly

Bridge, Poker

Imperfect information

Battleships, Stratego

Civilization, HoMM


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Game tree (2-player, deterministic, turns)

  • Concepts:

  • State: node in search space

  • Operator: valid move

  • Terminal test: game over

  • Utility function: value for outcome of the game

  • MAX: 1st player, maximizing its own utility

  • MIN: 2nd player, minimizing Max’s utility


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Minimax

  • Finding perfect play for deterministic games

  • Idea: choose move to position with highest minimax value = best achievable payoff against best play

  • E.g., 2-play game:



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Properties of minimax

  • Complete?

  • Optimal?

  • Time complexity?

    • b: branching factor

    • m: # moves in a game

Yes (if tree is finite)

Yes (against an optimal opponent)

O(bm)

  • For chess, b ≈ 35, m ≈100 for "reasonable" gamesTherefore, exact solution is infeasible


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Cutoff-test(state)

evaluationFunction(state)

Cutoff-test(state)

evaluationFunction(state)

Minimax algorithm with Imperfect Decisions


Evaluation function l.jpg

  • Chess

  • weight: Piece  Number

    • (w1) Pawn  1

    • (w2) Knight  3

    • (w3) Bishop  3

    • (w4) Rook  5

    • (w5) Queen  9

  • Function; state  Number

    • f1 = #(pawns,b)  #(pawns,w)

    • f2 = #(knight,b)  #(knight,w)

    • f3 = #(bishop,b)  #(bishop,w)

    • f4 = #(rook,b)  #(rook,w)

    • f5 = #(knight,b)  #(knight,w)

Evaluation Function

  • Evaluation Function

    • Is an estimate of the actual utility

    • Typically represented as a linear function:

      EF(state) = w1f1(state) + w2f2(state) + … + wnfn(state)

    • Example:


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Evaluation Function (2)

  • Obviously, the quality of the AI player depends on the evaluation function

  • Conditions for evaluation functions:

  • If n is a terminal node,

  • Computing EF should not take long

  • EF should reflect chances of winning

EF(n) = Utility(n)

If EF(state) > 3 then is almost-certain that blacks win








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Properties of α-β

  • 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)


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α is the value of the best (i.e., highest-value) choice found so far at any choice point along the path for max

If v is worse than α, max will avoid it

 prune that branch

Define β similarly for min

Why is it called α-β?


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The α-β algorithm found so far at any choice point along the path for


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The α-β algorithm found so far at any choice point along the path for


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Resource limits found so far at any choice point along the path for

Suppose we have 100 secs, explore 104 nodes/sec106nodes per move

Standard approach:

  • cutoff test:

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

  • evaluation function

    = estimated desirability of position


Evaluation functions l.jpg
Evaluation functions found so far at any choice point along the path for

  • 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 black queens), etc.


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Cutting off search found so far at any choice point along the path for

MinimaxCutoff is identical to MinimaxValue except

  • Terminal? is replaced by Cutoff?

  • Utility is replaced by Eval

    Does it work in practice?

    bm = 106, b=35  m=4

    4-ply lookahead is a hopeless chess player!

  • 4-ply ≈ human novice

  • 8-ply ≈ typical PC, human master

  • 12-ply ≈ Deep Blue, Kasparov


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Deterministic games in practice found so far at any choice point along the path for

  • Checkers: Chinook ended 40-year-reign of human world champion Marion Tinsley in 1994. Used a precomputed endgame database defining perfect play for all positions involving 8 or fewer pieces on the board, a total of 444 billion positions.

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

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

  • Go: human champions refuse to compete against computers, who are too bad. In go, b > 300, so most programs use pattern knowledge bases to suggest plausible moves.


Summary l.jpg
Summary found so far at any choice point along the path for

  • Games are fun to work on!

  • They illustrate several important points about AI

  • perfection is unattainable  must approximate

  • good idea to think about what to think about


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