A Skat Player Based on Monte Carlo Simulation Sebastian Kupferschmid and Malte Helmert Albert-Ludwigs-Universitat¨ Freiburg, Germany. fkupfersc,
[email protected] Abstract. We apply Monte Carlo simulation and alpha-beta search to the card game of Skat, which is similar to Bridge, but different enough to require some new algorithmic ideas besides the techniques developed for Bridge. Our Skat- playing program integrates well-known techniques such as move ordering with two new search enhancements. Quasi-symmetry reduction generalizes symme- try reductions, popularized by Ginsberg’s Partition Search algorithm, to search states which are “almost equivalent”. Adversarial heuristics generalize ideas from single-agent search algorithms like A∗ to two-player games, leading to guaran- teed lower and upper bounds for the score of a game position. Combining these techniques with state-of-the-art tree search algorithms, our program determines the game-theoretical value of a typical Skat hand (with perfect information) in 10 milliseconds. 1 Introduction Although mostly unknown in the English-speaking world, the game of Skat is the most popular card game in continental Europe, surpassed in world-wide popularity only by Bridge and Poker. With about 30 million casual players and about 40,000 people playing at a competitive level, Skat is mostly a German phenomenon, although national associ- ations exist in twenty countries on all six inhabited continents. It is widely considered the most interesting card game for three players. Despite its popularity, Skat has not been studied extensively by the AI community. This is not due to lack of challenge, as Skat is definitely a game of skill — significant experience is required to reach tournament playing strength — and all existing computer implementations play rather poorly.