game of chaos: 35e nederlands mathematisch congres utrecht; 19990408 the game of chaos peter

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.


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.


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 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


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


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


Lotteries

priceprob.

$31/3

$121/6

-$21/2

Expectation:1/2 . -2 + 1/6 . 12 + 1/3 . 3 = 2


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


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.


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


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


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

??

??


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.


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 !


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 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.


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


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



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



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


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


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

game of chaos: 35e nederlands mathematisch congres utrecht; 19990408 the game of chaos peter

Documents

game of chaos sorry

zerosum game

game gains

game trees root thorgrims

independent slide

urgats nodes

damaging spells

effects spells