Games solved:Now and in the future

1
Games solvedNow and in the future

• H.jaap van den Herik, Jos W.H.M. Uiterwijk, Jack
  van Rijswijck
• Summarized by Seung-joon Yi

2
Introduction

• The domain of strategic games
• To witness the intelligence of computers
• To establish the game-theoretic value of a game,
  i.e., the outcome when all participants play
  optimally.

3
Conventions

• Two-person zero-sum games with perfect
  information.
• Connection games
• Mancala games
• Endgame problems of chess-like games
• A subclass of other games.
• At least three different definitions of a
  solution can be used.
• Ultra-weakly solved
• game-theoretic value of the initial position has
  been determined
• Weakly solved
• For the initial position a strategy has been
  determined to achieve the game-theoretic value
  against any opposition.
• Strongly solved
• Such a strategy has been determined for all legal
  positions.

4
Classification by complexity

• Connect-four,Qubic,Nine mens morris, Go-moku is
  now solved.
• We can say that the predictions were rather
  accurate.

5
Classification by complexity

• State-space complexity
• No. of legal game positions reachable from the
  initial position of the game.
• Game-tree complexity
• No. of leaf nodes in the solution search tree of
  the initial position of the game.
• Categoy-1 is most suitible to being solved.
• Solving category-2 games is dependent on the
  computer power.
• Solving category-3 games is dependent on the
  development of new AI techniques.
• Solving category-4 games is practically
  impossible.

6
Questions

• Can perfect knowledge obtained from solved games
  be translated into rules and strategies which
  human beings can assimilate?
• Are such rules generic, or do they constitute a
  multitude of ad hoc recipes?
• Can methods be transferred between games? More
  specifically, are there generic methods for all
  category-n games, or is each game in a specific
  category a law unto itself?

7
Methods developed for solving games

• Brute-force methods
• Retrograde analysis
• For each position of some specific game or
  endgame the number of moves towards the best
  reachable goal is stored.
• Database is constructed by starting in terminal
  positions and then working backwards.
• Does not guarantee the best performance in a
  position which is game-theoretically drawn or
  lost.
• Enhanced transposition-table methods
• The traditional transposition table used in
  game-playing programs normally exploit the DEEP
  replacement scheme, i.e., when two different
  positions compete for the same entry in the
  table, the old position is overwritten by the
  newer one provided that the latter is searched at
  least as deep as the former.
• Research on this and other replacement schemes
  showed that two-level replacement schemes are
  often more appropriate.

8
Methods developed for solving games

• Knowledge-based methods
• Provides an appropriate move ordering or
  selection in the search trees.
• Threat-space search and ?-search
• Investigates whether by a sequence of threats, to
  which the opponent at any time has only a limited
  set of replies, a win can be forced.
• Proof-number search
• Depth-first proof-number search
• Pattern search

9
Classification by game type

• Convergence
• We consider a game to be convergent when the size
  of the state space decreases as the game
  prograsses.
• If the size of the state increases, the game is
  said to be divergent.
• Informally, convergent games start with many
  pieces on the board and pieces are gradually
  removed during the course of the game, while
  divergent games start with an empty or almost
  empty board and pieces are added during the game.

10
Convergent games

• Unlike divergent games, convergent games by their
  definition allow for the construction of endgame
  databases.
• These databases are a powerful tool for solving
  the games by retrograde analysis, starting at the
  goal states and working backward.
• Nine mens morris
• Mancala family of games
• Checkers and Chess endgames

11
Nine mens morris

• Gasser solved the game in 1995
• Total number of states7,673,759,269.
• He built all 28 w-b middle-game and endgame
  databases, and an 18-ply search from the start
  position using the DB was performed.
• After exloiting the DB, it was estabilished that
  the game-theoretic value of the game is a draw.
• Rules
• Opening phase
• Middle game phase
• Endgame phase

12
Mancala games

• All use a playing board with a number of holes,
  called pits.
• The pits are distributed over 2 or more rowsin
  addition there are 0,1,2 large holes, called
  stores.
• Usually starts with an equal number of stones in
  each pit, and with no stone in the stores.
• A turn consists of
• Taking all stones out of a pit
• Distributing them one after another over the
  subsequent pits.
• Hundreads of mancala games
• Awari
• Kalah
• Dakon

13
Awari

• At present, database for up to 38 stones are
  calculated and the 39-stone DB is under
  construction.
• 36-stone DB shows that Awari with 3 stones per
  pit in the initial position is a draw.
• Few more years at most is expected to solve the
  awari weakly.

14
Kalah

• Kalah(m,n), m being the number of pits per side
  and n the number of stones per pit, is weakly
  solved up to (6,5) using a combination of endgame
  databases and search.
• Standard game, Kalah(6,4), is a first-player win.

15
Dakon

• Winning opening
• If the starting player captures n(2n-1) stones,
  the game is over in a single move.
• Winning opening for Dakon-8 was found by hand
• Except for Dakon-3, many winning openings was
  found for n up to 18. Thus these games are weakly
  solved.

16
Checkers

• CHINOOK officially became the first man-machine
  world champion in any game in 1994.
• Next goalweakly solving the game of checkers for
  all 144 valid three-move opening sequences played
  in tournaments.
• Endgame databasesthe program has perfect
  information about all positions involving eight
  or fewer pieces on the board, a total of
  443,748,401,247 positions, compressed into 6GB.
• Middlegame databasesWhenever the programe is
  able to determine the game-theoretic value of a
  position during a game, the position is added to
  the middle-game database. In practice, position
  with as many as 22 pieces on the board have been
  solved.
• Verification of opening analysisstarting with
  published lines of play for each opening, deep
  searches are used to try and solve positions as
  close to the start of the game as possible. These
  solved positions are added to the middle-game
  database.
• CHINOOK proved the 100 year position for less
  than a second.

17
Chess endgames

• Endgames play a most prominent role in chess.
• Thompson built all important 5-piece endgames and
  made them accessible to the public.
• However it is unlikely that important endgames
  with more than 6 (non-blocked) pieces will be
  built in near future.

18
Summary-convergent games

• Nine mens Morris, the mancala games, and
  endgames in checkers and chess all belong to
  category 2.
• Endgame databases combined with search algorithm
  have been successful in attacking these games.
• Nine mens morris has been solved, as well as the
  smaller versions of Kalah and Dakon.
• Increasing computing power and storage capacity
  are expected to be instrumental in solving Awari
  within a few years, and Checkers within a
  decade.
• Full game of chess belongs to category 4.

19
Divergent games

• Divergent games are immune to the
  retrograde-analysis methods that were prevalent
  in section 2.
• Connection games
• Polymino games
• The games of Othello, Shogi, and Go.

20
Connection games

• Connect-four
• Weakly solved by allen using a brute-force
  approach, and by Allis using a knowledge-based
  approach.
• First-player win.
• Qubic
• Weakly solved in 1980, combining depth-first
  search and expert knowledge for ordering the
  moves.
• First-player win.
• Go-moku
• Allis estibalished that the game-theoretic value
  is a first-player win, using combination of
  threat-space search, proof-number search and
  database construction.

21
Connection games

• Renju
• Weakly solved as first-player win by Wagner and
  Virag in 2000, using transposition tables and
  threat-sequence search, and expert knowledge for
  no-threat moves.
• General k-in-a-row games
• K-in-a row in m by n boards
• Many game-theoretic values of many mnk-games have
  been published, based on using knowledge-based
  rules.

22
Polymino games

• Pentomino
• Two-player version has been weakly solved using a
  straightforward search program based on
  opening-move suggestions by the user.
• Domineering
• Many weakly-solved instances.

23
Othello and Shogi

• Othello
• Buros LOGISTELLO playes stronger than the human
  World Champion.
• 6 by 6 game is weakly solved.
• Shogi
• Not convergent, unlike western and chinese chess.
• Tsume-Shogi solving program SEO weakly solved a
  well-known problem with remarkable solution
  length of 1525 steps, setting a world record.

24
Go

• 19 by 19 Go is much too complex to solve with the
  current means.
• Localized endgame positions may be subject to
  exhaustivie analysis.
• Largest square board solved is 4 by 4, a
  first-player win.

25
Summary-nonconvergent games

• Connect-Four and Qubic belong to category 1
• Category 1 games are solved by both approaches.
• Go-moku, Renju,mnk-games to category 3.
• Category 3 games are solved by a combination of
  expert knowledge, threat-space search,
  threat-sequence search, proof-number search.
• The polyomino games, Othello, and Go on small
  boards belong to category 2.
• Category 2 games are solved by brute-force
  methods.
• Instances on larger boards will typicaly belong
  to category 4.

26
Game characteristics detemining solverbility

• Conclusion
• Low state-space complexity is more important than
  a low game-tree complexity as a determining
  factor in solving games.
• State-space complexity provides a bound on the
  complexity of games solvable by complete
  enumeration.
• The game-tree complexities form a real challenge
  for intelligent search methods.
• Brute-force versus Knowledge-based methods
• Games with relatively low state-space complexity
  have mainly been solved with brute-force emthods.
• Games with a relatively low game-tree complexity
  have mainly been solved with knowledge-based
  methods.
• Games with a relatively low complexity in both
  measure have been solved by both methods.

27
Conclusion

• The use of retrograde analysis in building DB has
  had the greatest impact on solving (end)games or
  almost-solving games.
• The knowledge-based methods mostly inform us the
  structure of the games.
• Questions revisited
• Can perfect knowledge obtained from solved games
  be translated into rules and strategies which
  human beings can assimilate?
• The most difficult solved (end)games are still a
  mystery for human experts.
• The best we can hope for is that perfect
  knowledge is translated into a correct strategy.
• Are such rules generic, or do they constitute a
  multitude of ad hoc recipes?
• In fact, the database itself is a long list of ad
  hoc recipes.
• Hence,the research question is how to combine
  them into tractable cluster of analogue positions
  and then to formulate a human-understandable
  rule.
• Can methods be transferred between games?
• Data mining attempt
• Understanding many intricacies of a game is a
  prerequisite to applying one of the proven
  methods out of the data-mining areas succesfully.