Projects • Analyze Existing Game project • Due next class. Re-check the guidelines on course’s website! • Create your Own Project • Due: Final game due Tuesday the last week of classes • 5-page Document: due Thursday the last week of classes • Award for Best Game by popular vote. • You received feedback • Do what we say and you will be exempt from final exam (meaning you get 100/100 score in it)
Game Theory • It is not a a theoretical approach to games • Game theory is the mathematical study of decision making • Origins in the field of economics • Examples of “games”: • Bankruptcy • Mutually Assured Destruction • It concerns with how to make decisions: • Idea: reason with the consequences of decisions (i.e., choosing a game action) made • Contrary to the book claim’s game theory techniques have been used to build successful game playing algorithms • Specifically for turn-based games
Turn-Based Strategy Games • Early strategy games was dominated by turn-based games • Derivated from board games • The Battle for Normandy (1982) • Nato Division Commanders (1985) • Panzer General (1994) • 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: • Mini-turns • Simultaneous: • Are Turn-based Strategy games dead? http://www.youtube.com/watch?v=jtJAO0222LI http://www.youtube.com/watch?v=E6xR4rDebtA
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 year • Fantasy-style (RPG) • Heroes of Might and Magic series. • Gambling 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? • … • 2005 Civilization IV • 2006 HoMM V
SideTrack: Game Interface Design: Contradicting Principles • Principle: All actions can be done from a single screen. • Classical example: Civilization • But: HoMM uses two interfaces: http://www.youtube.com/watch?v=Ed6suYzfckc
Game Trees • Two players (MAX and MIN) alternate moves (turn-based) • The tree indicating all possible movements by the players • The root of the tree indicates the starting situation • The children of the root indicate possible moves by MAX • The children of the children indicate possible moves by MIN • Tree continues alternating until it reaches end-game situations
… MIN moves (which depend on MAX’s last move) … First example MAX moves
Game Trees (2) • Concepts: • State: node in tree • Terminal node: game over • Utility function: value for outcome of the game. • MAX: 1st player, maximizing its own utility • MIN: 2nd player, minimizing Max’s utility Utility: 1 Max won 0 Max and Min draw -1 Max lost (Min won)
Minimax • Finding perfect play for deterministic, perfect information games • Idea: choose move to position with: highest utility for MAX = best achievable payoff against a rational opponent • Example: Utility is a number between 2 and 14
Properties of Minimax • Always find an optimal strategy for MAX (or MIN)? • Size of game tree? • b: branching factor (maximum number of moves a player can make in his/her turn) • m: # turns in an average game Yes (if tree is finite and opponent is rational) approximately bm • For Tic-Tac-Toe, b ≈ 9, m ≈10. Thus, the size of the tree is 910. Any computer nowadays will find the optimal strategy: • Always draw against a rational opponent • For chess, b ≈ 35, m ≈60. Thus, the size of the tree is 3560. Too large for computers today. Therefore tree can only be partially constructed. Look ahead of 6 turns for most computers
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, heuristics to guide construction of game tree with help of human grand masters • Othello: human champions refuse to compete against computers. Computers 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.
Saddle Points • “Of saddle points mindful you should be, young one hmmmmm” • Saddle point: Developing a strategy for playing the game that ensures victory (or non defeat) every time • Example: Tic-Tac-Toe • Results in loss of meaningful play • It is a theoretical possibility but in practice can’t figured it out is ok • But sometimes is emergent behavior • The Diablo II necromancer … … … Victory!
Summoning Necromancer http://www.youtube.com/watch?v=CTJzrQ8XrG0 • Some Design principles: • Rather weak character (not much of direct damage; can’t sustain a lot of damage) • Uses curses to weaken foe’s attacks/defenses • Killed monsters can be revived to serve the necromancer • But they last 2 minutes only (negative feedback!) • Reviving monsters, casting curses cost mana • Mana is limited and renews itself very slowly (negative feedback!) Saddle point: people killed shamans mobs which in could revive other mobs! (positive feedback)