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IMPERFECT INFORMATION GAMES looking for the right Model

© Games Workshop. © Games Workshop. IMPERFECT INFORMATION GAMES looking for the right Model. Algemeen Wiskunde Colloquium KdV-UvA Feb 27 2002. Peter van Emde Boas ILLC-FNWI-Univ. of Amsterdam Bronstee.com Software & Services B.V.

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IMPERFECT INFORMATION GAMES looking for the right Model

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  1. © Games Workshop © Games Workshop IMPERFECT INFORMATION GAMES looking for the right Model Algemeen Wiskunde Colloquium KdV-UvA Feb 27 2002 Peter van Emde Boas ILLC-FNWI-Univ. of Amsterdam Bronstee.com Software & Services B.V. References and slides available at: http://turing.science.uva.nl/~peter/teaching/thmod02.html

  2. Topics • Games, Computation and Computer Science • Game Representations • Decision Problems on Games • Backward Induction and PSPACE(The Holy Quadrinity) • The INIGMA Project

  3. First things First © Peter van Emde Boas © Peter van Emde Boas © Peter van Emde Boas A previous appearance ; Nov 21 1979 @ UvA others: Sep 25 1974, Mar 24 1982, Dec 04 1985

  4. First things First © Peter van Emde Boas Know thy Audience .....

  5. First things First © Peter van Emde Boas © Peter van Emde Boas The Horror of attending a talk..... ; Dec 13 1976 @ VUA

  6. GAMES AND COMPUTATION

  7. Games ??? Past (1980) position of Games in Mathematics & CS: Study object for a marginal part of AI (Chess playing programs) Recreational Mathematics (cf. Conway, Guy & Berlekamp Theory) Game Theory (Economy): von Neumann, Morgenstern, Aumann, Games in Logic: Determinacyin set theory, Ehrenfeucht-Fraise Games, ....

  8. Computer Science • Computation Theory • Complexity Theory • Machine Models • Algorithms • Knowledge Theory • Information Theory • Cryptography, Systems and Protocols WHERE ARE THE GAMES ?

  9. Games and Computer Science • Evasive Graph properties (1972-74) • Information & Uncertainty (Traub ea. - 1980+) • Pebble Game (Register Allocation, Theory 1970+) • Tiling Game (Reduction Theory - 1973+) • Alternating Computation Model (1977-81) • Interactive Proofs /Arthur Merlin Games (1983+) • Zero Knowledge Protocols (1984+) • Creating Cooperation on the Internet (1999+) • E-commerce (1999+) • Logic and Games (1950+) • Language Games, Argumentation (500 BC)

  10. Game Theory • Theory of Strategic Interaction • Attributes • Discrete vs. Continuous ( state space ) • Cooperative vs. Non-Cooperative ( pay-off ) • Perfect Information vs. Imperfect Information ( Information sets ); Knowledge Theory

  11. PARTICIPANTS & MOVES • Single player - no choices • Single player - random moves • Single player - choices : Solitaire • Two players - choices • Two players - choices and random moves • Two players - concurrent moves • More Players - Coalitions

  12. COMPUTATION • Deterministic • Nondeterministic • Probabilistic • Alternating • Interactive protocols • Concurrency

  13. COMPUTATION • Notion of Configurations: Nodes • Notion of Transitions: Edges • Non-uniqueness of transition: Out-degree > 1 - Nondeterminism • Initial Configuration : Root • Terminal Configuration : Leaf • Computation : Branch Tree • Acceptance Condition: Property of trees

  14. Linking Games and Computations • Single player - no choices : Routine : Determinism • Single player - choices : Solitaire (Puzzle) : Nondeterminism • Two players – choices : Finite Combinatorial Games : Alternating Computation • Single player - random moves : Gambling : Probabilistic Algorithms • Two players - choices and random moves : Interactive Proof Systems • Several players & Coalitions - group moves : Multi Prover Systems

  15. O S R -1/1 1/-1 2/0 -1/ 4 3/1 D 1/-1 -1/1 1 /-1 -3/ 2 1/ 4 5/-7 © Donald Duck 1999 # 35 GAME REPRESENTATIONS Strategic Format Game Graph Naive Format

  16. © Games Workshop © Games Workshop THORGRIM Dwarf High King URGAT Orc Big Boss Introducing the Opponents Games involve strategic interaction ......

  17. Game Trees (Extensive Form - close to Computation) Thorgrim’s turn Pay - offs 2/0 -1/ 4 Urgat’s turn Terminal node: 3/1 1 /-1 -3/ 2 1/ 4 5/-7 Non Zero-Sum Game: Pay-offs explicitly designated at terminal node Root

  18. Bi-Matrix Games O S R -1/1 1/-1 D 1/-1 -1/1 © Games Workshop © Games Workshop Ogre Squigg Runesmith Dragon © Games Workshop © Games Workshop © Games Workshop © Games Workshop A Game specified by describing the Pay-off Matrix ....

  19. Von Neumann’s Theorem O S R -1/1 1/-1 D 1/-1 -1/1 © Games Workshop © Games Workshop O S R D + )/2 ( + )/2: ( © Games Workshop © Games Workshop © Games Workshop © Games Workshop Mixed Strategy Nash Equilibrium; no player can improve his pay-off by deviation.

  20. © Donald Duck 1999 # 35 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 A Game specified by describing the rules of the game ....

  21. Questions about this Game • What if the number of matches is even? • Can any of the two players force a win by clever playing? • How does the winner depend on the number of matches • Is this dependency periodic? If so WHY?

  22. WHY WORRY ABOUT MODELS? Instance Size Instance Format Instances Question Algorithmic problem Space/Time Complexity Solutions Algorithm Machine Model The rules of the game called “Complexity Theory”

  23. 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 ??

  24. 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 Game Analysis in terms of Wood Measure. Wood Measure :configuration size & depth

  25. Decision Problems on Games from H.W. Lenstra: Aeternitatem Cogita

  26. 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?)

  27. 2/0 -1/ 4 3/1 2/0 1/-1 -3/ 2 1/ 4 5/-7 3/1 2/0 -1/ 4 -3/ 2 1/ 4 3/1 1/ 4 1 /-1 -3/ 2 1/ 4 5/-7 Backward Induction and its Complexity

  28. Backward Induction 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 -3/ 2 1/ 4 1/ 4 Non Zero-Sum Game: Pay-offs computed for all game nodes including the Root. Root

  29. Backward Induction 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

  30. 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: This Recursion proceeds in the game tree from the Leaves to the Root. Pitfalls: Draws possible? Terminating Game? Loops ?

  31. 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 ) Hence: Polynomial with Respect to the Wood Measure !

  32. 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.

  33. THE HOLY QUADRINITY FINITE COMBINATORIAL GAMES PSPACE UNRESTRICTEDUNIFORMPARALLELISM QUANTIFIED PROPOSITIONALLOGIC (QBF) : ALTERNATION

  34. Known Hardness results on Games in Complexity Theory(1980+) • QBF ( PSPACE ) ( the “mother game” ) • Tiling Games (NP, PSPACE,NEXPTIME,....) • Geography (PSPACE) • HEX (generalized or pure) (PSPACE) • Checkers, Go (PSPACE-hard) • Pebbling Game (PSPACE) (solitaire game!) • Block Moving Problems (PSPACE) • Chess (EXPTIME) (repetition of moves ! ) The Common View is that Games Characterize PSPACE

  35. The InIGMA Project O S R -1/1 1/-1 D 1/-1 -1/1 © Games Workshop © Games Workshop Ogre Squigg Runesmith Dragon © Games Workshop © Games Workshop © Games Workshop © Games Workshop

  36. 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?

  37. Matching Pennies L R l r L 1/ -1 -1/ 1 r l r l R -1/ 1 1/ -1 1/ -1 -1/ 1 1/ -1 -1/ 1 In the Game treeUrgat has a winning Strategy In the Matrix Formnobody has a winning strategy So Tree is incorrect representation of the game. Why ?

  38. INFORMATION SETS L R l r L 1/ -1 -1/ 1 r l r l R -1/ 1 1/ -1 1/ -1 -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.

  39. 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

  40. Modified combat of champs The squigg scares the dragon only after a sulfur bath..... ? W NW ? ? D R D R ? ? ? ? s s o s o o s o 1/ -1 1/ -1 1/ -1 1/ -1 -1/ 1 1/ -1 -1/ 1 -1/ 1 Backward Induction on Uniform Strategies

  41. Imperfect Information Version of the same game ? -1/ 1 W NW 1/ -1 -1/ 1 D R D R -1/ 1 -1/ 1 1/ -1 -1/ 1 s s o s o o s o 1/ -1 1/ -1 1/ -1 1/ -1 -1/ 1 1/ -1 -1/ 1 -1/ 1 ? W NW ? ? D R D R ? ? ? ? s s o s o o s o 1/ -1 1/ -1 1/ -1 1/ -1 -1/ 1 1/ -1 -1/ 1 -1/ 1

  42. 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...

  43. 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

  44. 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...)

  45. 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 PSPACE-hard.

  46. 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’swinning strategies correspond to input words outside L(R) , the language described by R; So Thorgrim wins the game iff L(R)  {0,1}*

  47. 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 qand input symbols s both players choose transitions...

  48. 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.....

  49. 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.

  50. 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).

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