Turn-Based Games

1. 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

2. 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

3. 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

4. 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 HoMMV • 2011 Civ V

5. 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)

6. 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

7. Deterministic Games • Every player’s actionresults in a single pre-determined state? • Yes: Deterministic game state action

8. 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 A B state1 (We don’t know apriori which until action is performed) state2 Action … Chance Games • Every player’s action results in a single pre-determined state? • No: Chance game

9. Perfect Information Games • Does the player knows all information about the current state of the game? • Yes: perfect information game

10. Imperfect Information Games • Does the player knows all information about the current state of the game? • No: imperfect information game

11. Types of AdversarialTBGs (from AI perspective) Chance Deterministic Chess, Go, rock-paper-scissors Perfect information Chutes and Ladders Bridge, Poker Imperfect information Warcraft 1 Civilization, HoMM 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

12. How Well does Computer Play Deterministic, Perfect Information Games? http://www.youtube.com/watch?v=NJarxpYyoFI

13. 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

14. 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:

15. Minimax algorithm

16. 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

17. Cutoff-test(state) evaluationFunction(state) Cutoff-test(state) evaluationFunction(state) Minimax algorithm with Imperfect Decisions

18. 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:

19. Example: Evaluation Function “all things been equal” White moves, Who is winning? Is this consistent with Evaluation function? Black Yes!

20. 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

21. 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: further explore states that may have more variance in results

22. Example: Horizon Problem “all things been equal” White moves, Who is winning? Is this consistent with Evaluation function? Black No!

23. 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

24. 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:

25. α-β pruning

26. α-β pruning example

27. α-β pruning example

28. α-β pruning example

29. α-β pruning example

30. 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, min will avoid it Therefore, prune that branch Principle of α-β Prunning

31. The α-β algorithm

32. The α-β algorithm

33. 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)

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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)

40. 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