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