IMPERFECT INFORMATION
GAMES; what makes them Hard
to Analyze ?
Amsterdam Aachen
Exchange-UvA
Feb 15 2002
© Games Workshop
© Games Workshop
Peter van Emde Boas
ILLC-FNWI-Univ. of Amsterdam
References and slides available at: http://turing.science.uva.nl/~peter/teaching/thmod02.html
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Topics
• Game Representations
• Forms of Backward Induction and
complexity
• Imperfect Information Games and
Jones’ example
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
GAME REPRESENTATIONS
O
2 / 0 -1 / 4
S
R
-1/1 1/-1
D
1/-1 -1/1
3/1
1 / -1
-3 / 2
1 / 4 5 / -7
© Donald Duck 1999 # 35
Strategic Format
Game Graph
Naive Format
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
WHY WORRY ABOUT MODELS?
Instance
Format
Instance
Size
Instances
Question
Algorithmic
problem
Solutions
Algorithm
Space/Time
Complexity
Machine
Model
The rules of the meta-game called “Complexity Theory”
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Introducing the Opponents
© Games Workshop
© Games Workshop
URGAT
THORGRIM
Orc Big Boss
Dwarf High King
Games involve strategic interaction ......
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Bi-Matrix Games
O
S
R
-1/1 1/-1
D
1/-1 -1/1
© Games Workshop
Runesmith
© Games Workshop
© Games Workshop
Dragon
© Games Workshop
Ogre
© Games Workshop
Squigg
© Games Workshop
A Game specified by describing
the Pay-off Matrix ....
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Game Trees
(Extensive Form - close to Computation)
Thorgrim’s turn
Pay - offs
2/0
-1 / 4
Terminal node:
1 / -1
Urgat’s turn
3/1
-3 / 2
1/4
Root
5 / -7
Non Zero-Sum Game:
Pay-offs explicitly
designated at terminal node
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
A Game
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
A Game specified by describing
the rules of the game ....
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Format and Input Size
Think about simple games like Tic-Tac-Toe
Naive size of the game indicated by measures like:
-- size configuration ( 9 cells possibly with marks)
-- depth (duration) game (at most 9 moves)
The full game tree is much larger : ~986410 nodes
Size of the strategic form beyond imagination.....
What size measure should we use for complexity
theory estimates ??
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
The Impact of the Format
The gap between the experienced size and the
size of the game tree is Exponential !
Another Exponential Gap between the
game tree and the strategic form.
These Gaps are highly relevant for Complexity!
The Challenge: Estimate Complexity of
Endgame Analysis in terms of experienced size.
Wood Measure : configuration size & depth
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Decision Problems on Games
• Which Player wins the game
– Winning Strategy ?
• End-game Analysis
• Termination of the Game
• Forcing States or Events
– Safety (no bad states)
– Lifeness (some good state will be reached)
• Power of Coalitions
• Game Equivalence (when are two games
the same?)
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Backward Induction and its
Complexity
2 / 0 -1 / 4
2 / 0 -1 / 4
3/1
1 / -1
-3 / 2
3/1
2/0
1 / 4 5 / -7
1 / -1
-3 / 2
1 / 4 5 / -7
3/1
-3 / 2
1/4
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
1/4
Backward Induction on trees
2 / 0 -1 / 4
3/1
2/0
1 / -1
-3 / 2
1 / 4 5 / -7
3/1
-3 / 2
1/4
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
At Probabilistic nodes: Pay-off evaluated by averaging
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Backward induction on Game Graphs
start
Initial labeling:
only final positions
are labeled.
start
T
T
U
U
U
U
D
D
T
D
T
T
T
Final labeling:
iterative apply BI rules
until no new nodes are
labeled. Remaining
nodes are Draw D
U
T
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Backward Induction in PSPACE?
The Standard Dynamic Programming Algorithm for
Backward Induction uses the entire Configuration
Graph as a Data Structure: Exponential Space !
Instead we can Use Recursion over Sequences of
Moves. So build a game tree for the game!
This Recursion proceeds in the game tree from the
Leaves to the Root.
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Backward Induction in PSPACE?
The Recursive scheme combines recursion
(over move sequence) with iteration (over locally
legal moves).
Space Consumption =
O( | Stackframe | . Recursion Depth )
| Stackframe | =
O( | Move sequence | + | Configuration| )
Recursion Depth = | Move sequence | =
O( Duration Game )
Thus Polynomial with Respect to the Wood Measure !
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
REASONABLE GAMES
Finite Perfect Information (Zero Sum)
Two Player Games
(possibly with probabilistic moves)
Structure: tree given by description,
where deciding properties like:
is p a position ?, is p final ? is p starting position ?,
who has to move in p ?,
generation of successors of p
are all trivial problems .....
The tree can be generated in time proportional to its size.....
Moreover the duration of a play is polynomial.
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Imperfect Information
Games
O
S
R
-1/1 1/-1
D
1/-1 -1/1
© Games Workshop
Runesmith
© Games Workshop
© Games Workshop
Dragon
© Games Workshop
Ogre
© Games Workshop
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Squigg
© Games Workshop
Imperfect Information makes
life more complex !
Examples of games where analyzing the
Perfect Information version is easier than the
Imperfect version.
Neil Jones produces such Example in 1978
I.E., perfect FAT in P and Imperfect IFAT
which is PSPACE hard......
How to compare two versions of a game?
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Combat of Champs ==
Matching Pennies
D
o
1 / -1
R
s
o
o
s
D
1 / -1
-1 / 1
R
-1 / 1
1 / -1
s
-1 / 1 -1 / 1 1 / -1
In the Game tree Urgat has a winning Strategy
In the Matrix Form nobody has a winning strategy
So Tree is incorrect representation of the game. Why ?
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
INFORMATION SETS
D
o
1 / -1
R
s
o
o
s
D
1 / -1
-1 / 1
R
-1 / 1
1 / -1
s
-1 / 1 -1 / 1 1 / -1
When Urgat has to Move he doesn’t know Thorgrim’s move.
Information sets capture this lack of Information.
Kripke style semantics.
Strategies must be Uniform
Urgat has no winning Uniform Strategy. Neither has Thorgrim
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Matrix Games are
Imperfect Information Games
Thorgrim’s Choice
of strategy
Urgat’s Choice
of strategy
Pay-off phase
Urgat doesn’t know the position he is in !
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Modified combat of champs
The squigg scares the dragon only after a sulfur bath.....
?
W
NW
?
?
D
?
o
1 / -1
D
R
?
s
o
s
-1 / 1 -1 / 1 1 / -1
R
?
o
1 / -1
?
s
o
s
1 / -1 -1 / 1 1 / -1
Backward Induction on Uniform Strategies
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Imperfect Information Version
of the same game ?
-1 / 1
W
-1 / 1
D
-1 / 1
R
o
s
1 / -1
-1 / 1
NW
-1 / 1
-1 / 1
D
1 / -1
s
o
1 / -1
R
o
s
1 / -1
1 / -1
1 / -1
-1 / 1
s
o
1 / -1
-1 / 1
?
W
?
?
o
1 / -1
D
R
s
-1 / 1
o
-1 / 1
NW
?
D
?
R
?
s
1 / -1
o
1 / -1
s
1 / -1
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
o
-1 / 1
?
s
1 / -1
Imperfect Information
makes life more complex !
Imperfect Information Game
<==>
Extension of Perfect Information Game Graph
with information sets and Uniform moves ???
Analysis remains in P !!
be it O(v.e) rather than O(v+e)
So something else is going on...
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Imperfect Information Games
Adaptation of BI on Graphs:
-- Simple games no longer are determinated
-- Information sets capture uncertainty
-- Uniform strategies are required
HOWEVER.....
-- Nodes may belong to multiple information
sets: disambiguation causes exponential
blow-up in size....
-- Earlier algorithms become incorrect if used
on nodes without disambiguation
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Neil Jones’ example (1978)
GAME FAT: Finite Automaton Traversal
Game played on (Deterministic) Finite Automaton
Some states are selected to be winning for Thorgrim
Players choose in turns an input symbol
(I.E. the next transition)
Just a pebble moving game on a game graph;
This can easily be analyzed in Polynomial time.
(even in linear time, if done efficiently...)
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Neil Jones’ example (1978)
GAME IFAT: Imperfect Finite Automaton
Traversal
Consider the version of FAT where Thorgrim doesn’t
observe Urgat’s moves:
Thorgrim can’t see where the pebble moves.
By a simple reduction from the problem to decide
whether a given regular expression describes the
language {0,1}* (shown to be PSPACE-complete by Meyer
and Stockmeyer) this version is proven to be PSPACEhard.
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Jones’ Reduction
For a given regular expression R first construct its NFA : M(R)
Next consider the following game:
Each turn Thorgrim chooses an input symbol: 0 or 1; next Urgat
chooses a legal transition in M(R) .
Thorgrim can’t observe the state of M(R) after the transition ...... !!!
Thorgrim decides when to end the game.
Urgat wins if an accepting state is reached at the end of the game;
otherwise Thorgrim wins the game
Thorgrim’s winning strategies correspond to input words outside
L(R) , the language described by R;
So Thorgrim wins the game iff L(R)  {0,1}*
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Jones’ Example ?
Question: in which sense is IFAT an
imperfect information version of FAT ?
Alternating choices between input symbols
and transitions is irrelevant difference;
introducing new states <q,s> for old states
q and input symbols s both players
choose transitions...
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Jones’ Example ?
What are the configurations in IFAT ??
in FAT the states in the FA are adequate representations
of the game configurations.
in IFAT the states are inadequate;
configurations are to be placed in an information set with
all other configurations where (according to Thorgrim)
the game could be...
and that depends on the input symbols processed so far.
Compare with subset construction for transforming an
NFA into a DFA. These subsets could be adequate.....
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Jones’ Example ?
These subsets could be adequate.....
SNAG: the subset construction increases the size
of the FA exponentially!
The jump of complexity from P to PSPACE is better
than we could have predicted; the naive graph based
backward induction yields an EXPTIME algorithm....
STILL: The subset construction does not yield the Kripke
model with Information sets.
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
What is the Kripke Model?
A candidate Kripke Model is the product
of the Automaton and its Deterministic
version obtained by the subset construction:
{<q, A> | q  A } with <q,A> ~ <q’,A> when
both q and q’  A .
Uniform strategies correspond to input
symbols (as should be the case).
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
The Punch line
Adding Imperfect Information in Jones’
example hardly increases the size of the
game in the Wood Measure, but increases
the game graph exponentially.
By coincidence, for the Perfect Information
version the wood measure and the size of the
game graph are proportional.
So again: Complexity with respect to
which measure.....???!!!
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
Conclusion
Imperfect Information Games can be
harder to analyze !!!
But doing the comparison is non trivial,
since it has everything to do with
(succinct) game representations
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.
CONCLUSIONS
© Morris & Goscinny
Peter van Emde Boas: Imperfect Information Games;
what makes them Hard to Analyze.