Peter van Emde Boas: Imperfect Information Games; looking for the right model. IMPERFECT INFORMATION GAMES looking for the right Model Peter van Emde Boas ILLC-FNWI-Univ. of Amsterdam Bronstee.com Software & Services B.V. References and slides available at: © Games Workshop Algemeen Wiskunde Colloquium KdV-UvA Feb

Peter van Emde Boas: Imperfect Information Games; looking for the right model. Topics Games, Computation and Computer Science Game Representations Decision Problems on Games Backward Induction and PSPACE (The Holy Quadrinity) The INIGMA Project

Peter van Emde Boas: Imperfect Information Games; looking for the right model. First things First © Peter van Emde Boas A previous appearance ; Nov 21 UvA others: Sep , Mar , Dec © Peter van Emde Boas

Peter van Emde Boas: Imperfect Information Games; looking for the right model. First things First Know thy Audience..... © Peter van Emde Boas

Peter van Emde Boas: Imperfect Information Games; looking for the right model. First things First © Peter van Emde Boas The Horror of attending a talk..... ; Dec 13 VUA

Peter van Emde Boas: Imperfect Information Games; looking for the right model. GAMES AND COMPUTATION

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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: Determinacy in set theory, Ehrenfeucht-Fraise Games,....

Peter van Emde Boas: Imperfect Information Games; looking for the right model. Computer Science Computation Theory Complexity Theory Machine Models Algorithms Knowledge Theory Information Theory Cryptography, Systems and Protocols

Peter van Emde Boas: Imperfect Information Games; looking for the right model. Evasive Graph properties ( ) Information & Uncertainty (Traub ea ) Pebble Game (Register Allocation, Theory 1970+) Tiling Game (Reduction Theory ) Alternating Computation Model ( ) 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) Games and Computer Science

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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

Peter van Emde Boas: Imperfect Information Games; looking for the right model. COMPUTATION Deterministic Nondeterministic Probabilistic Alternating Interactive protocols Concurrency

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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

Peter van Emde Boas: Imperfect Information Games; looking for the right model. GAME REPRESENTATIONS 2 / 02 / 0 5 / -71 / 4 -1 / 4 3 / 13 / 1 -3 / 21 / -1 R D OS -1/1 © Donald Duck 1999 # 35 Strategic Format Game Graph Naive Format

Peter van Emde Boas: Imperfect Information Games; looking for the right model. © Games Workshop URGAT Orc Big Boss © Games Workshop THORGRIM Dwarf High King Introducing the Opponents Games involve strategic interaction......

Peter van Emde Boas: Imperfect Information Games; looking for the right model. Extensive Form - close to Computation Game Trees (Extensive Form - close to Computation) Root Thorgrim’s turn Urgat’s turn Terminal node: Non Zero-Sum Game: Pay-offs explicitly designated at terminal node 2 / 02 / 0 5 / -71 / 4 -1 / 4 3 / 13 / 1 -3 / 21 / -1 Pay - offs

Peter van Emde Boas: Imperfect Information Games; looking for the right model. Bi-Matrix Games © Games Workshop Runesmith Dragon SquiggOgre R D OS 1/-1 -1/1 A Game specified by describing the Pay-off Matrix....

Peter van Emde Boas: Imperfect Information Games; looking for the right model. Von Neumann’s Theorem ( )/2 :+ ( )/2+ © Games Workshop R D SO R D OS 1/-1 -1/1 Mixed Strategy Nash Equilibrium; no player can improve his pay-off by deviation.

Peter van Emde Boas: Imperfect Information Games; looking for the right model. A Game © Donald Duck 1999 # 35 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....

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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?

Peter van Emde Boas: Imperfect Information Games; looking for the right model. WHY WORRY ABOUT MODELS? Algorithmic problem Instances Solutions Instance Format Question Instance Size Algorithm Space/Time Complexity The rules of the game called “Complexity Theory” Machine Model

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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 : ~ 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; looking for the right model. 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

Peter van Emde Boas: Imperfect Information Games; looking for the right model. Decision Problems on Games from H.W. Lenstra: Aeternitatem Cogita

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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; looking for the right model. Backward Induction and its Complexity 2 / 02 / 0 5 / -71 / 4 -1 / 4 3 / 13 / 1 -3 / 21 / -1 2 / 02 / 0 3 / 13 / 1 1 / 4 -3 / 2 1 / 4 2 / 02 / 0 5 / -71 / 4 -1 / 4 3 / 13 / 1 -3 / 21 / -1

Peter van Emde Boas: Imperfect Information Games; looking for the right model. Backward Induction Root Thorgrim’s turn Urgat’s turn Terminal node: Non Zero-Sum Game: Pay-offs computed for all game nodes including the Root. 2 / 02 / 0 5 / -71 / 4 -1 / 4 3 / 13 / 1 -3 / 21 / -1 2 / 02 / 0 3 / 13 / 1 1 / 4 -3 / 2 1 / 4

Peter van Emde Boas: Imperfect Information Games; looking for the right model. Backward Induction 2 / 02 / 0 5 / -71 / 4 -1 / 4 3 / 13 / 1 -3 / 21 / -1 2 / 02 / 0 3 / 13 / 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 At Probabilistic nodes: Pay-off evaluated by averaging

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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 ?

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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 !

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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; looking for the right model. THE HOLY QUADRINITY FINITE COMBINATORIAL GAMES QUANTIFIED PROPOSITIONAL LOGIC (QBF) : ALTERNATION PSPACE UNRESTRICTED UNIFORM PARALLELISM

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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

Peter van Emde Boas: Imperfect Information Games; looking for the right model. The InIGMA Project © Games Workshop Runesmith Dragon SquiggOgre R D OS 1/-1 -1/1

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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; looking for the right model. Matching Pennies L R lr 1 / / 1 LR 1 / -1-1 / 11 / -1 llr r -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; looking for the right model. INFORMATION SETS L R lr 1 / / 1 LR 1 / -1-1 / 11 / -1 llr r -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.

Peter van Emde Boas: Imperfect Information Games; looking for the right model. Urgat doesn’t know the position he is in ! Matrix Games are Imperfect Information Games Thorgrim’s Choice of strategy Urgat’s Choice of strategy Pay-off phase

Peter van Emde Boas: Imperfect Information Games; looking for the right model. Modified combat of champs DR 1 / -1 oos s -1 / 1 W NW 1 / -1-1 / 11 / -1 oos s -1 / 1 DR The squigg scares the dragon only after a sulfur bath / -1 ? ? ? ? ? ? Backward Induction on Uniform Strategies ?

Peter van Emde Boas: Imperfect Information Games; looking for the right model. Imperfect Information Version of the same game ? DR 1 / -1 oos s -1 / 1 W NW 1 / -1-1 / 11 / -1 oos s -1 / 1 DR 1 / -1 ? ? ? ? ? ? DR oos s -1 / 1 W NW 1 / -1-1 / 11 / -1 oos s -1 / 1 DR 1 / / 1 ?

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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; looking for the right model. 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; looking for the right model. Neil Jones’ example (1978) 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...) GAME FAT: Finite Automaton Traversal

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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; looking for the right model. 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 for old states q and input symbols s both players choose transitions...

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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; looking for the right model. 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; looking for the right model. 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 } with ~ when both q and q’  A. Uniform strategies correspond to input symbols (as should be the case).

Peter van Emde Boas: Imperfect Information Games; looking for the right model. 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; looking for the right model. 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; looking for the right model. CONCLUSIONS © Morris & Goscinny