GAMES AND COMPUTER SCIENCE Theoretical Models 1999 Peter van Emde Boas References available at: http://turing.wins.uva.nl/~peter/teaching/thmod99.html Most Papers will be made available in Library Games in Computer Science Information & Uncertainty (Traub ea.  80+)
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Theoretical Models 1999
Peter van Emde Boas
References available at: http://turing.wins.uva.nl/~peter/teaching/thmod99.html
Most Papers will be made available in Library
Combinatorial Analysis
Backward Induction
Number Theory
(Conway Guy Berlekamp)
Equilibria theory
(Nash)
Stochasitic Features
Optimization
Other names of importance:
Von Neumann & Morgenstern
Aumann
Shapley
Harsanyi
Starting with 15 matches
players alternatively take
1, 2 or 3 matches away until
none remain. The player
ending up with an odd
number of matches wins
the game
© Donald Duck 1999 # 35
Thorgrim’s turn
Terminal node:
Urgat looses
Urgat’s turn
Terminal node:
Thorgrim looses
Standard Interpretation:
Player unable to move
looses the game
Root
Thorgrim’s turn
T
U
Terminal node:
Urgat’s turn
T
T
U
U
T
Terminal node:
Free Interpretation:
Winner explicitly
designated at terminal node
Root
Thorgrim’s turn
2/0
1/ 4
Terminal node:
Urgat’s turn
3/1
1/1
3/ 2
1/ 4
5/7
Terminal node:
Non ZeroSum Game:
Payoffs explicitly
designated at terminal node
Root
SUBGAME
Thorgrim’s turn
T
U
Terminal node:
Urgat’s turn
T
T
U
U
T
Terminal node:
Free Interpretation:
Winner explicitly
designated at terminal node
Root
Thorgrim’s turn
T
U
Terminal node:
Urgat’s turn
T
T
T
U
U
T
T
Terminal node:
U
U
Free Interpretation:
Winner explicitly
designated at terminal node
U
Root
Thorgrim’s turn
2/0
1/ 4
Terminal node:
Urgat’s turn
3/1
2/0
1/1
3/ 2
1/ 4
5/7
3/1
Terminal node:
3/ 2
1/ 4
Non ZeroSum Game:
Payoffs explicitly
designated at terminal node
1/ 4
Root
T
U
2/0
1/ 4
T
3/1
T
2/0
U
T
U
T
1/1
3/ 2
1/ 4
5/7
T
3/1
U
U
3/ 2
1/ 4
U
1/ 4
At terminal nodes: Payoff as explicitly given
At Thorgrim’s nodes: Payoff inherited from Thorgrim’s optimal choice
At Urgat’s nodes: Payoff inherited from Urgat’s optimal choice
For strictly competetive games this is the MaxMin rule
Extension used:
Thorgrim wins if he has
an odd number when the
game terminates.
This allows for even n .
Relevant feature: parity of number of matches collected
so far (not the number itself!)
Four types of configurations remain:
T/E : Thorgrim has to play and has an even number
T/O : Thorgrim has to play and has an odd number
U/E : Urgat plays, while Thorgrim has an even number
U/O : Urgat plays, while Thorgrim has an odd number
nU / EU / OT / ET / O
18UUT / 1T / 2
17 U T T / 1 U
16 U T U T / 3
15 U U T / 2T / 3
14 U U T / 2T / 1
13 TU U T / 1
12 TU T / 3 U
11 U U T / 3 T / 2
10UUT / 1T / 2
9 U T T / 1 U
8 U T U T / 3
7 U U T / 2T / 3
6 U U T / 2T / 1
5 TU U T / 1
4 TU T / 3 U
3 U U T / 3 T / 2
2UUT / 1T / 2
1 U T T / 1 U
0 U T U T
Configuration Type
Existential
Universal
Negating
+

+
Accepting


+
+

+

Rejecting
Computation Tree
+
Configuration Type
+


+
+
Existential
+

Universal
+
+
+



Negating

+

+
+

+
Accepting


+
+

+

Rejecting
Evaluation Full Computation Tree
This Tree Accepts
Infinite Branches ?
Requires third quality :Indeterminatenodes
Universalnode isindeterminateiff it has no rejecting
son and at least oneindeterminateson
Existentialnode isindeterminateiff it has noaccepting
son and at least oneindeterminateone
Negatingnode isindeterminateiff its son is
Infinite Branches ?
Universalnode isacceptingiff it has no rejecting
son and noindeterminateson (all sons are accepting)
Existentialnode isacceptingiff it has oneaccepting
Son; indeterminateand rejecting sons don’t matter
Negatingnode isacceptingiff its son is rejecting
Requires RecursiveEvaluation of computation tree !
+

^
The proper way of
Recursive evaluation ???
+

^
^
+

+

^
+

^
….
….
….
Indeterminate : ^
+

^
Recursive evaluation ==
Solving LEAST FIXEDPOINT EQUATION !
+

^
+

Partial order ≤
of definedness
Extends to functions
defined on the tree:
F ≤ G iff "x[F(x) ≤ G(x)]
^
+

^
+

^
+

^
….
….
….
OK
NOK
SET U with partial order ≤ and
least element ^
Countable chains have least upper bounds
DOMAIN :=
x0≤ x1 ≤ x2 ≤ ….. ≤ xn ≤ xn+1 ≤ …. > xw=:i xi
"i[xi ≤ xw] and "i[xi ≤ y] ==> xw ≤ y
FUNCTION F which is:
MONOTONE: x ≤ y ==> F(x) ≤ F(y)
CONTINUOUS: F( i xi ) = iF(xi)
OPERATOR :=
THEOREM: If F is an operator defined over domain U
then the equation X = F( X ) has a least solution W .
This solution is obtained as the limit of the sequence of
iterates: ^ ≤ F( ^ ) ≤ F(F( ^ )) ≤ ….
W = iFi ( ^ )
APPLICATION: U := domain of evaluations of tree
F := single application of recursive rule