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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. - PowerPoint PPT Presentation
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Turn-Based Games
Héctor Muñoz-Avila
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
Turn-Based Strategy GamesEarly 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
• At least 3 sub-styles are very popular:– “Civilization”-style games
• Civilization V will be released soon
– Fantasy-style (RPG)• Heroes of Might and Magic series
– Poker games• Poker Academy
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?• …• 2006 HoMM V• 2011 Civ V
Coming back: How to Construct Good AI?
• Idea: Lets just use A* and define a good heuristic for the game
Search space: a bipartite treeAfter 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)
Categories of Games from an Artificial Intelligence (AI) Perspective
• Categories were, in part, the result of pursuing to apply AI techniques to construct “better” computer-controlled opponents
• Depending on the categories, building such AI turns out to be easier or more difficult
• In the context of game design, provides another way to analyze games – Specifically a different way from the traditional
classification of games by genre
Deterministic Games• Every player’s action results in a single pre-determined state?
Yes: Deterministic game
state
action
Chance Games• Every player’s action results in a single pre-determined state?
No: Chance game
A
B
Action: A cast a damage spell on B
Outcome:• B blocks spell with
20% chances• If B does not block
spell, then damage dealt to B is randomly choose between 25%-40% of player’s B health points
Action
state1
state2
…
(We don’t know apriori which until action is performed)
Perfect Information Games
• Does the player knows all information about the current state of the game?Yes: perfect information game
Imperfect Information Games
• Does the player knows all information about the current state of the game?No: imperfect information game
Types of AdversarialTBGs (from AI perspective)
Perfect information
Imperfect information
Deterministic Chance
Chess, Go, rock-paper-scissors
Warcraft 1
Chutes and Ladders
Civilization, HoMMBridge, Poker
We are going to study solutions for perfect information, deterministic games, for groundbreaking research including chance elements and imperfect information come to invited lecture on Thursday STEPS 101, 4PM
How Well does Computer Play Deterministic, Perfect Information Games?
http://www.youtube.com/watch?v=NJarxpYyoFI
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
• 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:
Minimax algorithm
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
evaluationFunction(state)
evaluationFunction(state)
Cutoff-test(state)
Cutoff-test(state)
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:
Chess• weight: Piece Number
(w1) Pawn 1 (w2) Knight 3(w3) Bishop 3(w4) Rook 5 (w5) Queen 9
• Function; state Numberf1 = #(pawns,w) #(pawns,b)f2 = #(knight,w) #(knight,b)f3 = #(bishop,w) #(bishop,b)f4 = #(rook,w) #(rook,b)f5 = #(knight,w) #(knight,b)
Example: Evaluation Function
“all things been equal”White moves,Who is winning?
Is this consistent with Evaluation function?
Black
Yes!
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 longEF should reflect chances of winning
EF(n) = Utility(n)
If EF(state) > 3 then is almost-certain that blacks win
Cutting Off Search
• When to cutoff minimax expansion?
• Potential problem with cutting off search: Horizon problem
• Solution:
Fixed depth limitIterative deepening until times runs out
Decision made by opponent is damaging but cannot be “seen” because of cutoff
Quiescent: further explore states that may have more variance in results
Example: Horizon Problem
“all things been equal”White moves,Who is winning?
Is this consistent with Evaluation function?
Black
No!
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 master12-turns ≈ Deep Blue, Kasparov
• How to look ahead more than 4 turns? Use α-β pruning
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:
α-β pruning
α-β pruning example
α-β pruning example
α-β pruning example
α-β pruning example
Principle of α-β 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
oTherefore, prune that branch
• β is the lowest-value found so far at any choice point along the path for min If v , min will avoid it
oTherefore, prune that branch
The α-β algorithm
The α-β algorithm
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)
Note
• The next 5 slides are from David W. Aha (NRL) presentation at Lehigh University in Fall’04
• Claim: search space in Civilization is much larger than Chess
Example Game: FreeCiv(Chance, adversarial, imperfect information game)
http://www.freeciv.org
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
FreeCiv ScenarioGeneral 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
FreeCiv ConceptsConcepts in an Initial Knowledge Base• Resources: Collection and use
o Food, production, trade (money)• Terrain:
o Resources gained per turno Movement requirements
• Units:o Type (Military, trade, diplomatic, settlers, explorers)o Healtho Combat: Offense & defenseo 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
o Population Growtho Happinesso Pollution
• Civilizations (e.g., military strength, aggressiveness, finances, cities, units)• Diplomatic states & negotiations
FreeCiv DecisionsCivilization 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
Unit decisions
Diplomacy 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
• Task (e.g., where to build a city, whether/where to engage in combat, espionage)• Movement
• Whether to sign a proffered peace treaty with another civilization• Whether to offer a gift
FreeCiv CP Decision SpaceVariables • Civilization-wide variables
o N: Number of civilizations encounteredo D: Number of diplomatic states (that you can have with an opponent)o G: Number of government types available to youo R: Number of research advances that can be pursued o I: Number of partitions of income into entertainment, money, & research
• U: #Unitso L: Number of locations a unit can move to in a turn
• C: #Citieso Z: Number of citizens per cityo S: Citizen status (i.e., laborer, entertainer, doctor)o 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
o This becomes large with the number of units and citieso Example: N=3; D=5; G=3; R=4; I=10; U=25; L=4; C=8; Z=10; S=3; B=10o Size of decision space (i.e., possible next states): 2.5*1065 (in one turn!)
o Comparison: Decision space of chess per turn is well below 140 (e.g., 20 at first move)
Open Discussion: Learning versus Exhaustive Search
• Learns from experience• Learns same solution
versus minimax “rational” opponent
• Explores all possibilities• Generates best solution
versus “rational” opponent