Peter van Emde Boas: Imperfect Information Games; what makes them Hard to Analyze. IMPERFECT INFORMATION GAMES; what makes them Hard to Analyze ? Peter.

Peter van Emde Boas: Imperfect Information Games; what makes them Hard to Analyze.

IMPERFECT INFORMATION GAMES; what makes them Hard

to Analyze ?

Peter van Emde Boas

ILLC-FNWI-Univ. of Amsterdam

References and slides available at: http://turing.science.uva.nl/~peter/teaching/thmod02.html

© Games Workshop© Games Workshop

Amsterdam Aachen Exchange-UvA

Feb 15 2002


Topics

• Game Representations

• Forms of Backward Induction and complexity

• Imperfect Information Games and Jones’ example


GAME REPRESENTATIONSGAME REPRESENTATIONS

2 / 0

5 / -71 / 4

-1 / 4

3 / 1

-3 / 21 / -1R

D

O S

1/-1

1/-1

-1/1

-1/1

© Donald Duck 1999 # 35

Strategic Format Game Graph Naive Format


WHY WORRY ABOUT MODELS?

Algorithmic problem

Instances

Solutions

InstanceFormat

Question

Instance Size

Algorithm Space/TimeComplexity

The rules of the meta-game called “Complexity Theory”

MachineModel


© Games Workshop

URGATOrc Big Boss

© Games Workshop

THORGRIMDwarf High King

Introducing the Opponents

Games involve strategic interaction ......Games involve strategic interaction ......


Bi-Matrix Games

© Games Workshop © Games Workshop© Games Workshop© Games Workshop


Runesmith Dragon SquiggOgre

R

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O S

1/-1

1/-1

-1/1

-1/1

A Game specified by describing A Game specified by describing the Pay-off Matrix ....the Pay-off Matrix ....


Game Trees (Extensive Form - close to ComputationExtensive Form - close to Computation)

Root

Thorgrim’s turn

Urgat’s turnTerminal node:

Non Zero-Sum Game:Pay-offs explicitly designated at terminal node

2 / 0

5 / -71 / 4

-1 / 4

3 / 1

-3 / 21 / -1

Pay - offs


A Game

© Donald Duck 1999 # 35

Starting with 15 matchesplayers alternatively take1, 2 or 3 matches away untilnone remain. The playerending up with an oddnumber of matches winsthe game

A Game specified by describing A Game specified by describing the rules of the game ....the rules of the game ....


Format and Input SizeThink about simple games like Tic-Tac-ToeNaive 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 nodesSize of the strategic form beyond imagination.....

What size measure should we use for complexitytheory estimates ??


The Impact of the FormatThe 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


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?)


Backward Induction and its Backward Induction and its ComplexityComplexity

2 / 0

5 / -71 / 4

-1 / 4

3 / 1

-3 / 21 / -1

2 / 0

3 / 1

1 / 4-3 / 2

1 / 4

2 / 0

5 / -71 / 4

-1 / 4

3 / 1

-3 / 21 / -1


Backward Induction on trees

2 / 0

5 / -71 / 4

-1 / 4

3 / 1

-3 / 21 / -1

2 / 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 choiceAt Urgat’s nodes: Pay-off inherited from Urgat’s optimal choiceAt Probabilistic nodes: Pay-off evaluated by averaging


Backward induction on Game Graphsstart

TU

Initial labeling:only final positionsare labeled.

start

TU

Final labeling:iterative apply BI rulesuntil no new nodes arelabeled. Remaining nodes are Draw D

D

T

U

U

U

T

T

TT

D D


Backward Induction in PSPACE?

The Standard Dynamic Programming Algorithm forBackward Induction uses the entire ConfigurationGraph 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 theLeaves to the Root.


Backward Induction in PSPACE?The Recursive scheme combines recursion(over move sequence) with iteration (over locallylegal 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 !


REASONABLE GAMESFinite 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.


Imperfect Information Imperfect Information GamesGames

© Games Workshop © Games Workshop© Games Workshop© Games Workshop


Runesmith Dragon SquiggOgre

R

D

O S

1/-1

1/-1

-1/1

-1/1


Imperfect Information makes life more complex !

Examples of games where analyzing thePerfect Information version is easier than theImperfect version.

Neil Jones produces such Example in 1978I.E., perfect FAT in P and Imperfect IFATwhich is PSPACE hard......

How to compare two versions of a game?


Combat of Champs ==Matching Pennies

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o s

1 / -1

1 / -1-1 / 1

-1 / 1

D R

1 / -1 -1 / 1 1 / -1

o os s

-1 / 1

In the Game tree Urgat has a winning StrategyIn the Matrix Form nobody has a winning strategy

So Tree is incorrect representation of the game. Why ?


INFORMATION SETS

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 UniformUrgat has no winning Uniform Strategy. Neither has Thorgrim

D

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o s

1 / -1

1 / -1-1 / 1

-1 / 1

D R

1 / -1 -1 / 1 1 / -1

o os s

-1 / 1


Urgat doesn’t know the position he is in !

Matrix Games areImperfect Information Games

Thorgrim’s Choice of strategy

Urgat’s Choice of strategy

Pay-off phase


Modified combat of champs

D R

1 / -1 1 / -1

o os s

-1 / 1

W NW

1 / -1 -1 / 1 1 / -1

o os s

-1 / 1

D R

The squigg scares the dragon only after a sulfur bath.....

1 / -1

?

?

?

?

?

?

Backward Induction on Uniform Strategies

?


Imperfect Information Version of the same game ?

D R

1 / -1 1 / -1

o os s

-1 / 1

W NW

1 / -1 -1 / 1 1 / -1

o os s

-1 / 1

D R

1 / -1

?

?

?

?

?

?

D R1 / -1

1 / -1 1 / -1

o os s

-1 / 1

W NW

1 / -1 -1 / 1 1 / -1

o os s

-1 / 1

D R

1 / -1

1 / -1

-1 / 1 -1 / 1

-1 / 1

-1 / 1

-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...


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


Neil Jones’ example (1978)

Game played on (Deterministic) Finite Automaton

Some states are selected to be winning for ThorgrimPlayers 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...)

GAME FAT: Finite Automaton Traversal


Neil Jones’ example (1978)

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 PSPACE-hard.

GAME IFAT: Imperfect Finite Automaton Traversal


Jones’ ReductionFor 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 outsideL(R) , the language described by R;So Thorgrim wins the game iff L(R) {0,1}*


Jones’ Example ?Question: in which sense is IFAT animperfect information version of FAT ?

Alternating choices between input symbols and transitions is irrelevant difference;introducing new states <q,s> for old statesq and input symbols s both players choose transitions...


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 anNFA into a DFA. These subsets could be adequate.....


Jones’ Example ?These subsets could be adequate.....

SNAG: the subset construction increases the sizeof the FA exponentially!

The jump of complexity from P to PSPACE is betterthan we could have predicted; the naive graph basedbackward induction yields an EXPTIME algorithm....

STILL: The subset construction does not yield the Kripkemodel with Information sets.


What is the Kripke Model?

A candidate Kripke Model is the productof the Automaton and its Deterministicversion obtained by the subset construction:

{<q, A> | q A } with <q,A> ~ <q’,A> whenboth q and q’ A .Uniform strategies correspond to input symbols (as should be the case).


The Punch lineAdding Imperfect Information in Jones’example hardly increases the size of thegame in the Wood Measure, but increasesthe game graph exponentially.

By coincidence, for the Perfect Informationversion the wood measure and the size of thegame graph are proportional.

So again: Complexity with respect towhich measure.....???!!!


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


CONCLUSIONS

© Morris & Goscinny

Peter van Emde Boas: Imperfect Information Games; what makes them Hard to Analyze. IMPERFECT INFORMATION GAMES; what makes them Hard to Analyze ? Peter.

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