INTELLIGENT SYSTEM FOR PLAYING TAROK Mitja Luštrek & Matjaž Gams Jožef Stefan Institute Ljubljana, Slovenia

PERFECT AND IMPERFECT INFORMATION GAMES Perfect information (players have full knowledge of the state of the game) Chess, backgammon Checkers, Othello Connect-four ... Imperfect information (players have only partial knowledge of the state of the game) Bridge Poker Tarok

THE GAME – TAROK Very popular in Central Europe Many variants (tarock, taroky, königsrufen...) Three players: two against one 54 cards: suits and trumps – taroks The objective is winning tricks

THE PROGRAM – SILICON TAROKIST Tarok-playing programs exist, but little is known of how they work. Tarok.net (www.tarok.net) Tarock World (www.gatecentral.com/triangle) ... We developed Silicon Tarokist. Freely available (tarok.bocosoft.com) Plays reasonably well as judged by human players.

GAME TREE SEARCH Alpha-beta algorithm is used to search a single game tree. Nodes – game states Edges – moves

SAMPLING Monte Carlo sampling is used to generate samples of other players’ hands.

ALPHA-BETA ENHANCEMENTS Transposition table Fuzzy transposition table Similar to partition search (bridge program GIB, M. L. Ginsberg, 1996) Move ordering Adjusting the width of search window Pruning the game tree

TRANSPOSITION TABLE Usually: transposition table stores single game states and their values. Partition search: for each encountered game state, a set of states with equal value is calculated and stored together with the value. Silicon Tarokist: the set of equivalent game states is determined heuristically.

OTHER ALPHA-BETA ENHANCEMENTS Move ordering Moves that cause cut-offs should be tried first. History heuristic: moves that have caused cut-offs in previously searched game states are given priority. Adjusting the width of search window Narrower search window causes more cut-offs, thus speeding up the search. Minimal window search: non-first children of a node are searched with minimal window, since we are trying to show they are inferior to the first one. Pruning the game tree Some moves can be discarded because they are either clearly bad or redundant – the same effect can be achieved by another move.

MONTE CARLO SAMPLING ENHANCEMENT Monte Carlo sampling has demonstrable deficiencies. Nevertheless, it works. Deficiency we observed: An assumption about the state of the game is made. Sequence of bad, but inevitable move – good move is evaluated equally as good move – bad, but inevitable move. Sometimes bad, but inevitable move is made first. Then it turns out it is not inevitable. Solution: In addition to full search, search to the depth of one trick is performed. This emphasizes immediate profit. A combination of both searches is used for the final decision.

RESULTS Game tree search algorithm in Silicon Tarokist searches 184-times less nodes than alpha-beta using uses 86-times less time. The program does not play flawlessly, but it is a challenging opponent. For truly high-level play, game tree search that we use in inadequate. It is too shallow for long-term strategies to be developed. It will either have to be improved significantly or another – probably knowledge-based – way to develop long-term strategies will have to be devised.