Turn-Based Games

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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 Schaeffers AAW 05 presentation
• My own

Héctor Muñoz-Avila
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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 didnt 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 Maxs 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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Minimax algorithm
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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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Minimax algorithm with Imperfect Decisions
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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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Example Evaluation Function
all things been equal White moves, Who is
winning? Is this consistent with Evaluation
function?
Black
Yes!
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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) gt 3 then is almost-certain that
blacks win
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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

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Example Horizon Problem
all things been equal White moves, Who is
winning? Is this consistent with Evaluation
function?
Black
No!
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a-ß 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 a-ß
  pruning

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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

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a-ß pruning
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a-ß pruning example
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a-ß pruning example
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a-ß pruning example
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a-ß pruning example
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Principle of a-ß Prunning

• a 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 ? a, max will avoid it
• Therefore, prune that branch
• ß is the lowest-value found so far at any choice
  point along the path for min
• If v ? a, min will avoid it
• Therefore, prune that branch

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The a-ß algorithm
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The a-ß algorithm
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Properties of a-ß

• 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)

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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 gt 300, so
  most programs use pattern knowledge bases to
  suggest plausible moves.

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Additional Notes

• The next 5 slides are form David W. Aha (NRL)
  presentation at Lehigh University in Fall04

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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
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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

Game Option Y1 Y2 Y3
World size Small Normal Large
Difficulty level Warlord (2/6) Prince (3/6) King (4/6)
Opponent civilizations 5 5 7
Level of barbarian activity Low Medium High
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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

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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

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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(DNGRILU(SZB)C) this ignores both
  other variables and domain knowledge
• This becomes large with the number of units and
  cities
• Example N3 D5 G3 R4 I10 U25 L4
  C8 Z10 S3 B10
• Size of decision space (i.e., possible next
  states) 2.51065 (in one turn!)
• Comparison Decision space of chess per turn is
  well below 140 (e.g., 20 at first move)