1. 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 Amsterdam Aachen Exchange-UvA Feb 15 2002

2. 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

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

4. Peter van Emde Boas: Imperfect Information Games; what makes them Hard to Analyze. WHY WORRY ABOUT MODELS? Algorithmic problem Instances Solutions Instance Format Question Instance Size Algorithm Space/Time Complexity The rules of the meta-game called “Complexity Theory” Machine Model

5. Peter van Emde Boas: Imperfect Information Games; what makes them Hard to Analyze. © Games Workshop URGAT Orc Big Boss © Games Workshop THORGRIM Dwarf High King Introducing the Opponents Games involve strategic interaction......

6. Peter van Emde Boas: Imperfect Information Games; what makes them Hard to Analyze. Bi-Matrix Games © Games Workshop Runesmith Dragon SquiggOgre R D OS 1/-1 -1/1 A Game specified by describing the Pay-off Matrix....

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

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

9. 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 ??

10. 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

11. 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?)

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

13. Peter van Emde Boas: Imperfect Information Games; what makes them Hard to Analyze. Backward Induction on trees 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

14. Peter van Emde Boas: Imperfect Information Games; what makes them Hard to Analyze. Backward induction on Game Graphs start T U Initial labeling: only final positions are labeled. start T U Final labeling: iterative apply BI rules until no new nodes are labeled. Remaining nodes are Draw D D T U U U T T T T D D

15. 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.

16. 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 !

17. 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.

18. Peter van Emde Boas: Imperfect Information Games; what makes them Hard to Analyze. Imperfect Information Games © Games Workshop Runesmith Dragon SquiggOgre R D OS 1/-1 -1/1

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

20. Peter van Emde Boas: Imperfect Information Games; what makes them Hard to Analyze. Combat of Champs == Matching Pennies D R os 1 / -1 -1 / 1 DR 1 / -1-1 / 11 / -1 oos s -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 ?

21. Peter van Emde Boas: Imperfect Information Games; what makes them Hard to Analyze. 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 Uniform Urgat has no winning Uniform Strategy. Neither has Thorgrim D R os 1 / -1 -1 / 1 DR 1 / -1-1 / 11 / -1 oos s -1 / 1

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

23. Peter van Emde Boas: Imperfect Information Games; what makes them Hard to Analyze. 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 / -1 ? ? ? ? ? ? Backward Induction on Uniform Strategies ?

24. Peter van Emde Boas: Imperfect Information Games; what makes them Hard to Analyze. 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 -1 / 1 ?

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

26. 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

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

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

29. 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}*

30. 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 for old states q and input symbols s both players choose transitions...

31. 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.....

32. 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.

33. 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 } with ~ when both q and q’  A. Uniform strategies correspond to input symbols (as should be the case).

34. 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.....???!!!

35. 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

36. Peter van Emde Boas: Imperfect Information Games; what makes them Hard to Analyze. CONCLUSIONS © Morris & Goscinny