game of chaos: 35e nederlands mathematisch congres utrecht; 19990408 the game of chaos peter
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Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
The Game of Chaos
Peter van Emde BoasILLC-WINS-UvA
Plantage Muidergracht 241018 TV [email protected]
Evert van Emde BoasLord Trevor Productions
Franz Lisztlaan 52102 CJ Heemstede
35e Nederlands Mathematisch CongresUtrecht 19990408
© Wizards of the Coast, inc.
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
The Game of ChaosSorry: it is a French CardGame of ChaosSorcery
Play head or tails against atarget opponent. The looserof the game looses one life.The winner of the game gainsone life, and may choose torepeat the procedure. For everyrepetition the ante in life isdoubled.© Wizards of the Coast, inc.
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Magic; the GatheringCustomizable card game: build a deck using a very large collectionof 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 effectsDefense: Blocking attacking creatures, protecting spells and effectsSpells require Mana obtained by tapping lands or activating otherMana 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.....
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Game Trees
Root
Thorgrim’s turn
Urgat’s turn
Terminal node
Non Zero-Sum Game:Payoffs explicitly designated at terminal node
2 / 0
5 / -71 / 4
-1 / 4
3 / 1
-3 / 21 / -1
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Backward Induction
2 / 0
5 / -71 / 4
-1 / 4
3 / 1
-3 / 21 / -12 / 0
3 / 1
1 / 4-3 / 2
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
T
TU
U
T
UT T
T
UU
U
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
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
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Lotteries
priceprob.
$31/3
$121/6
-$21/2
Expectation:1/2 . -2 + 1/6 . 12 + 1/3 . 3 = 2
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Compound Lottery
priceprob.
$31/3
$121/6
-$21/2
$31/2
-$21/2
1/5 4/5
priceprob.
$37/15
$121/30
-$21/2
In compound lotteries all drawings are assumed to be independent
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Flipping a coin
HEADS TAILS
1 / -1 -1/ 1 -1 / 1 1 / -1
h ht t1/21/2 1/21/2
Expectation 0 / 0 0 / 0
Thorgrim calls head or tails and Urgat flips the coin. Urgat’s move is irrelevant. Nature determines the outcome.
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
The Game Tree
0
1 -1
3 -31-1
7 -1 3 -5 5 -3 1 -7
7 -9 11 -5 5 -11 9 -7
33 / -3
1/2 1/2
Denotes
X
Y
Y
XThorgrim and Urgat bothstart with 5 lives
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
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
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Comparing Complex LotteriesAllais Example
0 1 00.01 0.89 0.10
0.89 0.11 00.9 0 0.1
$0M $1M $5M $0M $1M $5M
$0M $1M $5M $0M $1M $5M
??
??
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Von Neumann-Morgenstern Utility
Rational Players may be assumed to maximize theexpectation of Something.
Let’s call this Something Utility.
Works nice for 2-outcome Lotteries: Something = chance of winning.
So let’s reduce the n-outcome Lotteries to 2-outcomeCompound Lotteries:
Each intermediate outcome is “equivalent” to a suitable 2-outcome Lottery. The involved chance
determines the Utility.
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Utility Intermediate Outcome
W L
p 1-ppp :=
u(L) = au(W) = bu(D) = x
a < bD
Lot-1 Lot-3
EELot-1( u ) = p.b + (1-p).a EELot-3( u ) = x
If p is large (almost 1) : Lot-1 > Lot-3For p small (almost 0) : Lot-1 < Lot-3
So for some intermediate p, say q: Lot-1 ≈ Lot-3
qq ≈ Lot-3 whence u(D) = q.b + (1-q).a !
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Utility Lottery = Expected Utility Outcomes
p1 pn
1 ni
pi p1 pnpi
W L
qi 1-qi
W L
piqi 1- piqi
≈
u(W) = 1 , u(L) = 0 , u(i) = qi
piqi = u(Lot-3) = piu( i ) = EE Lot-1 u(outcome)
Lot-1 Lot-2 Lot-3
≈
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Game of Chaos3
3 / -31/2 1/2
Denotes
X
Y
Y
X
Structure of the game tree independent of the choice of the utilities.
uT,1: uT,1(n) = nuT,2: uT,2(n) = if n ≥ vopp then 1 elif n ≤ - vself then -1 else 0 fi
uU,1: uU,1(n) = -nuU,2: u U,2(n) = if n ≥ vself then - 1 elif n ≤ - vopp then 1 else 0 fi
© Wizards of the Coast, inc.
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Linear Utilities0/0
-1/11/-1
3/-3
7/-7
7/-7
-1/1 1/-1 -3/3
-3/3-1/1
-9/9
3/-3
11/-11
5/-5
5/-5
1/-1-7/7
-7/7-11/11 9/-9
-5/5
-5/5
Both Thorgrim and Urgat use utility u1
Thorgrim and Urgat bothstart with 5 lives
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Go for the Kill!0
1 -1
3 -31-1
7 -1 3 -5 5 -3 1 -7
7 -9 11 -5 5 -11 9 -7
1/-1
1/-11/-1 1/-1 1/-1
1/-10/0
0/0
0/0
0/0
0/0 0/0 0/0
.5/-.5 .5/-.5-.5/.5 -.5/.5
-1/1 -1/1 -1/1 -1/1
-1/1 -1/1
Both Thorgrim and Urgat use utility u2
Thorgrim and Urgat bothstart with 5 lives
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Mixed Utilities0
1 -1
3 -31-1
7 -1 3 -5 5 -3 1 -7
7 -9 11 -5 5 -11 9 -7
1/-7
1/-111/-7 1/-5 1/-9
1/-50/1
0/-1
0/0
0/1
0/-3 0/3 0/-1
.5/-3 .5/-1-.5/1 -.5/3
-1/9 -1/5 -1/11 -1/7
-1/5 -1/7
Thorgrim uses u2 ; Urgat uses u1
Thorgrim and Urgat bothstart with 5 lives
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Winning is all0
1 -1
3 -31-1
7 -1 3 -5 5 -3 1 -7
7 -9 11 -5 5 -11 9 -7
1/0
1/01/0 1/0 1/0
1/0.5/.5
.75/.25 .75/.25.25/.75 .25/.75
0/1 0/1 0/1 0/1
0/1 0/1
Utilities: Thorgrim uses u3,T: u3.T(n) = if n ≥ vopp then 1 else 0 fi Urgat uses u3,U: u3.U(n) = if - n ≥ vopp then 1 else 0 fi
.5/.5.5/.5
.5/.5
.5/.5 .5/.5
.5/.5Thorgrim and Urgat bothstart with 5 lives
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Unequal Start0
1 -1
3 -31-1
7 -1 3 -5 5 -3 1 -7
7 -9 11 -5 5 -11 9 -7
1/-1
1/-1
1/-1
1/-1 1/-1
1/-10/0
.25/-.25 0/0
0/0
0/0 0/0
.5/-.5 .5/-.5
-.5/.5
-.5/.5
-1/1
-1/1 -1/1
-1/1
Thorgrim: 6 livesUrgat: 4 livesutilities used u2
3 -13
-2111 17 -13
-1/1-1/1
1/-1
1/-1
-1/1
0/0
.5/-.5
0/0
.125/-.125
Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
Thorgrim’s last stand
0
1
-1
3 -1
7 -1
Thorgrim: 1 liveUrgat: 6 lives
Utilities: Thorgrim uses u3: u3(n) = if n ≥ vopp then 1 else 0 fi Urgat uses u2
1/-1 0/1
.5/00/1
0/1.25/.5
.125/.75