Presented at Board Game Studies, Brugge, 2011. @inproceedings{teytaud:inria-00625794, hal_id = {inria-00625794}, url = {http://hal.inria.fr/inria-00625794}, title = {{Lemmas on Partial Observation, with Application to Phantom Games}}, author = {Teytaud, Fabien and Teytaud, Olivier}, abstract = {{Solving games is usual in the fully observable case. The partially observable case is much more difficult; whenever the number of strategies is finite (which is not necessarily the case, even when the state space is finite), the main tool for the exact solving is the construction of the full matrix game and its solving by linear programming. We here propose tools for approximating the value of partially observable games. The lemmas are relatively general, and we apply them for deriving rigorous bounds on the Nash equilibrium of phantom-tic-tac-toe and phantom-Go.}}, language = {Anglais}, affiliation = {Laboratoire de Recherche en Informatique - LRI , TAO - INRIA Saclay - Ile de France}, booktitle = {{Computational Intelligence and Games}}, address = {Seoul, Cor{\'e}e, R{\'e}publique Populaire D{\'e}mocratique De}, audience = {internationale }, year = {2011}, month = Sep, pdf = {http://hal.inria.fr/inria-00625794/PDF/phantomatari.pdf}, }

1. Difficult Endgame Analysis
Fabien Teytaud, Olivier Teytaud

2. Outline
Fast state
of the art
 Computer-Games
 Endgame analysis in board games
 Endgame analysis in partially observable games
Our results
 Discussion

3. Computer games
[Le damier de l'opéra]

4. How strong computers are ?
 In Chess ?
 In Go ?
 In Backgammon ?
 In poker ?
 ...

5. Endgame analysis in board
games
 In Chess :
 Nalimov tables
 Five pieces or fewer has been completly analyzed
 Six pieces has been completly analyzed except trivial
positions
 Just few 7 pieces positions have been analyzed
 Perfect play

6. Endgame analysis in board
games
[Søren Riis in Bielefeld 2010]

7. Endgame analysis in board
games
[Søren Riis in Bielefeld 2010]

8. Endgame analysis in partially
observable games
New challenge ?

9. Endgame analysis in partially
observable games
 What is phantom games ?

10. Endgame analysis in partially
observable games
 What are phantom games ?
 No information about the moves of the
opponent.
 Some examples :
 In Chess :
 Kriegspiel
 Dark Chess
 In Go :
 Phantom Go

11. Endgame analysis in partially
observable games
 What is phantom games ?
 No information about the moves of the
opponent.
 Some examples :
 In Chess :
 Kriegspiel
 Dark Chess
 In Go :
 Phantom Go

12. Endgame analysis in partially
observable games
 What is phantom games ?
 Example of a Dark Chess position

13. Endgame analysis in partially
observable games
 What are phantom games ?
 Example of a Dark Chess position
[Wikipedia]

14. Algorithms
 Retrograde propagation no more possible
 Worst case analysis :
 Findsperfect strategy against an opponent
who sees the board
 Does not evaluate properly real positions
(example in next slide)

15. A Phantom-Go example
 Let's take this position :
 Black to play
 Position is done so that White wins if and only
if he saves all his groups

16. A Phantom-Go example

17. Analysis in standard Go ?
 Easy win for White. Each time black
attacks a group then White responds in
the same group and make two eyes

18. Analysis in standard Go ?

19. Analysis in standard Go ?

20. Analysis in standard Go ?

21. Analysis in standard Go ?

22. Analysis in standard Go ?

23. But in Phantom-Go ?
 White does not know which group is under attack
 Then, White can win only by playing in the exactly in
the right order
 Probably that this happens : 1/8 !
 Which algorithm for this case ?
 D.Auger :D. Auger. Multiple tree for partially observable
Monte-Carlo Tree Search. EvoGames, LNCS, Springer
 Adapted to Dark Chess : no observation until game over

24. Dark Chess examples
 Small board sizes
 3x3
 4x4
 8x8?
 Only King vs King final

25. Dark Chess examples

26. Dark Chess results

27. Dark Chess results

28. Dark Chess results

29. Dark Chess results

30. Conclusion
 Usual methods are approximations of perfect
play
 We propose an (asymptotically) exact
algorithm
 If
finite horizon
 But general case undecidable (Teytaud, 2010)
 Tractable in some cases
 First ever real endgame analysis in PO
games

31. Dark Chess results