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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) Jonathan Schaeffer’s AAW 05 presentation My own H é ctor Mu ñ oz-Avila Turn-Based Strategy Games

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turn based games

Turn-Based Games

  • sources:
  • http://www.game-research.com/
  • www.gamespot.com
  • Wikipedia.org
  • Russell & Norvig AI Book; Chapter 5 (and slides)
  • Jonathan Schaeffer’s AAW 05 presentation
  • My own

Héctor Muñoz-Avila

turn based strategy games
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
turn based games continues to be a popular game genre
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
some historical highlights
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
side tracking game design contradicting principles
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
coming back how to construct good ai
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)
types of adversarial tbgs from ai perspective
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

game tree 2 player deterministic turns
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
minimax
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:
properties of minimax
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
minimax algorithm with imperfect decisions

Cutoff-test(state)

evaluationFunction(state)

Cutoff-test(state)

evaluationFunction(state)

Minimax algorithm with Imperfect Decisions
evaluation function

Chess

  • weight: Piece  Number
    • (w1) Pawn  1
    • (w2) Knight  3
    • (w3) Bishop  3
    • (w4) Rook  5
    • (w5) Queen  9
  • Function; state  Number
    • f1 = #(pawns,w)  #(pawns,b)
    • f2 = #(knight,w)  #(knight,b)
    • f3 = #(bishop,w)  #(bishop,b)
    • f4 = #(rook,w)  #(rook,b)
    • f5 = #(knight,w)  #(knight,b)
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:
example evaluation function
Example: Evaluation Function

“all things been equal”

White moves,

Who is winning?

Is this consistent with Evaluation function?

Black

Yes!

evaluation function 2
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

cutting off search
Cutting Off Search
  • When to cutoff minimax expansion?
  • Potential problem with cutting off search: Horizon problem
  • Solution:
  • Fixed depth limit
  • Iterative deepening until times runs out
  • Decision made by opponent is damaging but cannot be “seen” because of cutoff
  • Quiescent: states that are unlikely to exhibit wild swings in the values of the evaluation functions
example horizon problem
Example: Horizon Problem

“all things been equal”

White moves,

Who is winning?

Is this consistent with Evaluation function?

Black

No!

pruning motivation
α-β pruning: Motivation
  • A good program may search 1000 positions per second
  • In a chess tournament, a player gets 150 seconds per move
  • Therefore, the program can explore 150,000 positions per move
  • With a branching factor of 34, this will mean a look ahead of 3 or 4 moves
  • Facts:
  • 4-turns ≈ human novice
    • 8-turns ≈ typical PC, human master
    • 12-turns ≈ Deep Blue, Kasparov
  • How to look ahead more than 4 turns? Use α-β pruning
example
Example:
  • 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:
principle of prunning
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 α, max will avoid it

Therefore, prune that branch

β is the lowest-value found so far at any choice point along the path for min

If v α, min will avoid it

Therefore, prune that branch

Principle of α-β Prunning
properties of
Properties of α-β
  • Pruning preserves completeness and optimality of original minimax algorithm
  • Good move ordering improves effectiveness of pruning
  • With "perfect ordering," time complexity = O(bm/2)

Therefore, doubles depth of search

  • Used in PC games today (9 moves look-ahead, Grand Master level)
deterministic games in practice
Deterministic games in practice
  • 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, 24 processors, quiescent identified with help of human grand masters
  • 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.
additional notes
Additional Notes
  • The next 5 slides are form David W. Aha (NRL) presentation at Lehigh University in Fall’04
example game freeciv chance adversarial imperfect information game
Example Game: FreeCiv(Chance, adversarial, imperfect information game)

Civilization II(MicroProse)

  • Civilization II (1996-): 850K+ copies sold
    • PC Gamer: Game of the Year Award winner
    • Many other awards
  • Civilization series (1991-): Introduced the civilization-based game genre

FreeCiv (Civ II clone)

  • Open source freeware
  • Discrete strategy game
  • Goal: Defeat opponents, or build a spaceship
  • Resource management
    • Economy, diplomacy, science, cities, buildings, world wonders
    • Units (e.g., for combat)
  • Up to 7 opponent civs
  • Partial observability

http://www.freeciv.org

freeciv scenario
FreeCiv Scenario

General description

  • Game initialization: Your only unit, a “settler”, is placed randomly on a random world (see Game Options below). Players cyclically alternate play
  • Objective: Obtain highest score, conquer all opponents, or build first spaceship
  • Scoring: “Basic” goal is to obtain 1000 points. Game options affect the score.
    • Citizens: 2 pts per happy citizen, 1 per content citizen
    • Advances: 20 pts per World Wonder, 5 per “futuristic” advance
    • Peace: 3 pts per turn of world peace (no wars or combat)
    • Pollution: -10pts per square currently polluted
  • Top-level tasks (to achieve a high score):
    • Develop an economy
    • Increase population
    • Pursue research advances
    • Opponent interactions: Diplomacy and defense/combat
freeciv concepts
FreeCiv Concepts

Concepts in an Initial Knowledge Base

  • Resources: Collection and use
    • Food, production, trade (money)
  • Terrain:
    • Resources gained per turn
    • Movement requirements
  • Units:
    • Type (Military, trade, diplomatic, settlers, explorers)
    • Health
    • Combat: Offense & defense
    • Movement constraints (e.g., Land, sea, air)
  • Government Types (e.g., anarchy, despotism, monarchy, democracy)
  • Research network: Identifies constraints on what can be studied at any time
  • Buildings (e.g., cost, capabilities)
  • Cities
    • Population Growth
    • Happiness
    • Pollution
  • Civilizations (e.g., military strength, aggressiveness, finances, cities, units)
  • Diplomatic states & negotiations
freeciv decisions
FreeCiv Decisions

Civilization decisions

  • Choice of government type (e.g., democracy)
  • Distribution of income devoted to research, entertainment, and wealth goals
  • Strategic decisions affecting other decisions (e.g., coordinated unit movement for trade)

City decisions

  • Production choice (i.e., what to create, including city buildings and units)
  • Citizen roles (e.g., laborers, entertainers, or specialists), and laborer placement
    • Note: Locations vary in their terrain, which generate different amounts of food, income, and production capability

Unit decisions

  • Task (e.g., where to build a city, whether/where to engage in combat, espionage)
  • Movement

Diplomacy decisions

  • Whether to sign a proffered peace treaty with another civilization
  • Whether to offer a gift
freeciv cp decision space
FreeCiv CP Decision Space

Variables

  • Civilization-wide variables
    • N: Number of civilizations encountered
    • D: Number of diplomatic states (that you can have with an opponent)
    • G: Number of government types available to you
    • R: Number of research advances that can be pursued
    • I: Number of partitions of income into entertainment, money, & research
  • U: #Units
    • L: Number of locations a unit can move to in a turn
  • C: #Cities
    • Z: Number of citizens per city
    • S: Citizen status (i.e., laborer, entertainer, doctor)
    • B: Number of choices for city production

Decision complexity per turn (for a typical game state)

  • O(DNGRI*LU*(SZB)C) ; this ignores both other variables and domain knowledge
    • This becomes large with the number of units and cities
    • Example: N=3; D=5; G=3; R=4; I=10; U=25; L=4; C=8; Z=10; S=3; B=10
    • Size of decision space (i.e., possible next states): 2.5*1065 (in one turn!)
  • Comparison: Decision space of chess per turn is well below 140 (e.g., 20 at first move)