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Game of Chaos: 35e Nederlands Mathematisch Congres Utrecht; 19990408
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