1 / 22

The Game of Chaos

The Game of Chaos. Peter van Emde Boas ILLC-WINS-UvA Plantage Muidergracht 24 1018 TV Amsterdam peter@wins.uva.nl Evert van Emde Boas Lord Trevor Productions Franz Lisztlaan 5 2102 CJ Heemstede 35e Nederlands Mathematisch Congres Utrecht 19990408. © Wizards of the Coast, inc.

eve
Download Presentation

The Game of Chaos

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Game of Chaos Peter van Emde Boas ILLC-WINS-UvA Plantage Muidergracht 24 1018 TV Amsterdam peter@wins.uva.nl Evert van Emde Boas Lord Trevor Productions Franz Lisztlaan 5 2102 CJ Heemstede 35e Nederlands Mathematisch Congres Utrecht 19990408 © Wizards of the Coast, inc.

  2. The Game of Chaos Sorry: it is a French Card Game of Chaos Sorcery Play head or tails against a target opponent. The looser of the game looses one life. The winner of the game gains one life, and may choose to repeat the procedure. For every repetition the ante in life is doubled. © Wizards of the Coast, inc.

  3. Magic; the Gathering Customizable card game: build a deck using a very large collection of available cards. Both players start out with 20 lives. Number of lives ≤ 0 means you have lost the duel. Move = playing land, casting a spell, combat, .... Attack: summoning creatures, damaging spells, damaging effects Defense: Blocking attacking creatures, protecting spells and effects Spells require Mana obtained by tapping lands or activating other Mana sources. Mana exists in 5 colors and a generic variant. Spells exist in the same 5 colors or a generic variant (artifacts) For almost every rule in the game there exists a card creating an exception against it when successfully cast.....

  4. Game Trees Non Zero-Sum Game: Payoffs explicitly designated at terminal node Thorgrim’s turn Urgat’s turn Root Terminal node 1/ 4 5 /-7 1 /-1 -3 / 2 3 /1 2 /0 -1 / 4

  5. Backward Induction 1/ 4 U U 1/ 4 U -3 / 2 T 3 /1 U T T U 1/ 4 5 /-7 T 1 /-1 -3 / 2 2 /0 T 3 /1 T U 2 /0 -1 / 4 At terminal nodes: Pay-off as explicitly given At Thorgrim’s nodes: Pay-off inherited from Thorgrim’s optimal choice At Urgat’s nodes: Pay-off inherited from Urgat’s optimal choice For strictly competitive games this is the Max-Min rule

  6. CHANCE MOVES • Chance moves controlled by another player (Nature) who is not interested in the result • Nature is bound to choose his moves fairly with respect to commonly known probabilities • Resulting outcomes for active players become lotteries

  7. Lotteries price prob. -$2 1/2 $12 1/6 $3 1/3 Expectation: 1/2 . -2 + 1/6 . 12 + 1/3 . 3 = 2

  8. price prob. -$2 1/2 $3 1/2 -$2 1/2 $12 1/6 $3 1/3 Compound Lottery 1/5 4/5 price prob. -$2 1/2 $12 1/30 $3 7/15 In compound lotteries all drawings are assumed to be independent

  9. HEADS TAILS h t h t 1 / -1 -1/ 1 -1 / 1 1 / -1 Flipping a coin 1/2 1/2 1/2 1/2 Expectation 0/ 0 0/ 0 Thorgrim calls head or tails and Urgat flips the coin. Urgat’s move is irrelevant. Nature determines the outcome.

  10. The Game Tree 3 Denotes 3 / -3 1/2 1/2 X Y Thorgrim and Urgat both start with 5 lives X Y 0 1 -1 -1 -3 3 1 7 -1 3 -5 5 -3 1 -7 7 -9 11 -5 5 -11 9 -7

  11. WHY UTILITY FUNCTIONS? • Backward Induction is based on preferences rather than numbers • Numbers as a tool for expressing preferences works OK when chance moves are absent • We like to compute expected pay-off at chance nodes. • Expected pay-off is sensitive to scaling • Comparing complex lotteries is non-trivial

  12. ?? 0 1 0 0.01 0.89 0.10 $0M $1M $5M $0M $1M $5M ?? 0.9 0 0.1 0.89 0.11 0 $0M $1M $5M $0M $1M $5M Comparing Complex LotteriesAllais Example

  13. Von Neumann-Morgenstern Utility Rational Players may be assumed to maximize the expectation of Something. Let’s call this SomethingUtility. Works nice for 2-outcome Lotteries: Something = chance of winning. So let’s reduce the n-outcome Lotteries to 2-outcome Compound Lotteries: Each intermediate outcome is “equivalent” to a suitable 2-outcome Lottery. The involved chance determines the Utility.

  14. p 1-p W L Utility Intermediate Outcome Lot-1 Lot-3 u(L) = a u(W) = b u(D) = x a<b p := D ELot-1( u ) = p.b + (1-p).a ELot-3( u ) = x If p is large (almost 1) : Lot-1 > Lot-3 For psmall (almost 0) : Lot-1 < Lot-3 So for some intermediate p, say q: Lot-1 ≈Lot-3 q ≈ Lot-3 whence u(D) = q.b + (1-q).a !

  15. Utility Lottery = Expected Utility Outcomes Lot-1 Lot-2 Lot-3 S piqi 1- S piqi p1 pi pn p1 pi pn ≈ ≈ w1 wi wn W L qi 1-qi W L u(W) = 1 , u(L) = 0 , u(wi) = qi Spiqi= u(Lot-3) = Spiu(wi) = E Lot-1 u(outcome)

  16. Game of Chaos 3 Denotes 3 / -3 1/2 1/2 X Y © Wizards of the Coast, inc. X Y Structure of the game tree independent of the choice of the utilities. uT,1: uT,1(n) = n uT,2: uT,2(n) = if n ≥ vopp then 1 elif n ≤ - vselfthen -1 else 0 fi uU,1: uU,1(n) = -n uU,2: u U,2(n) = if n ≥ vself then - 1 elif n ≤ - vopp then 1 else 0 fi

  17. Linear Utilities Thorgrim and Urgat both start with 5 lives 0/0 1/-1 -1/1 -1/1 3/-3 1/-1 -3/3 7/-7 -1/1 3/-3 -3/3 1/-1 -5/5 5/-5 -7/7 7/-7 -9/9 11/-11 -5/5 5/-5 -11/11 9/-9 -7/7 Both Thorgrim and Urgat use utility u1

  18. Go for the Kill! Thorgrim and Urgat both start with 5 lives 0 0/0 1 -1 0/0 0/0 -1 -3 3 1 -.5/.5 .5/-.5 .5/-.5 -.5/.5 7 -1 3 -5 5 -3 1 -7 0/0 0/0 0/0 0/0 1/-1 -1/1 1/-1 -1/1 7 -9 11 -5 5 -11 9 -7 1/-1 -1/1 1/-1 -1/1 1/-1 -1/1 1/-1 -1/1 Both Thorgrim and Urgat use utility u2

  19. Mixed Utilities Thorgrim and Urgat both start with 5 lives 0 0/0 1 -1 0/-1 0/1 -1 -3 3 1 -.5/1 .5/-1 .5/-3 -.5/3 7 -1 3 -5 5 -3 1 -7 0/1 0/-3 0/3 0/-1 1/-7 -1/5 1/-5 -1/7 7 -9 11 -5 5 -11 9 -7 1/-7 -1/9 1/-11 -1/5 1/-5 -1/11 1/-9 -1/7 Thorgrim uses u2 ; Urgat uses u1

  20. Winning is all Thorgrim and Urgat both start with 5 lives 0 .5/.5 1 -1 .5/.5 .5/.5 -1 -3 3 1 .25/.75 .75/.25 .75/.25 .25/.75 7 -1 3 -5 5 -3 1 -7 .5/.5 .5/.5 .5/.5 1/0 0/1 1/0 .5/.5 0/1 7 -9 11 -5 5 -11 9 -7 1/0 0/1 1/0 0/1 1/0 0/1 1/0 0/1 Utilities: Thorgrim uses u3,T: u3.T(n) = if n ≥ voppthen 1 else 0 fi Urgat uses u3,U: u3.U(n) = if - n ≥ voppthen 1 else 0 fi

  21. Unequal Start Thorgrim: 6 lives Urgat: 4 lives utilities used u2 .125/-.125 0 1 -1 0/0 .25/-.25 -1 -3 3 1 .5/-.5 .5/-.5 0/0 -.5/.5 7 -1 3 -5 5 -3 1 -7 0/0 .5/-.5 -.5/.5 0/0 0/0 1/-1 1/-1 -1/1 7 -9 11 -5 3 -13 5 -11 9 -7 1/-1 -1/1 1/-1 0/0 0/0 -1/1 1/-1 -1/1 1/-1 -1/1 11 -21 17 -13 1/-1 -1/1 1/-1 -1/1

  22. Thorgrim’s last stand Thorgrim: 1 live Urgat: 6 lives .125/.75 0 -1 1 .25/.5 0/1 -1 3 .5/0 0/1 7 -1 1/-1 0/1 Utilities: Thorgrim uses u3: u3(n) = if n ≥ voppthen 1 else 0 fi Urgat uses u2

More Related