On Pruning Techniques for Multi-Player Games

Max (Luckhardt and Irani, 1986) is the extension of the minimax backup rule to multi-player games. We have shown that only a limited version of alpha-beta pruning, shallow pruning, can be applied to a max search tree. We extend this work by calculating the exact bounds needed to use this pruning technique. In addition, we show that branch-and-bound pruning, using a monotonic heuristic, has the same limitations as alpha-beta pruning in a max tree. We present a hybrid of these algorithms, alpha-beta branch-and-bound pruning, which combines a monotonic heuristic and backed-up values to prune even more effectively. We also briefly discuss the reduction of a n-player game to a ‘paranoid’ 2-player game. In Sergeant Major, a 3-player card game, we averaged node expansions over 200 height 15 trees. Shallow pruning and branch-and-bound each reduced node expansions by a factor of about 100. Alpha-beta branch-and-bound reduced the expansions by an additional factor of 19. The 2-player reduction was a factor of 3 better than alpha-beta branchand-bound. Using heuristic bounds in the 2-player reduction reduced node expansions another factor of 12. Introduction and Overview Much work and attention has been focused on two-player games and alpha-beta minimax search (Knuth, Moore, 1975). This is the fundamental technique used by computers to play at the championship level in games such as chess and checkers. Alpha-beta pruning works particularly well on games of two players, or games with two teams, such as bridge. Much less work has been focused on games with three or more teams or players, such as Hearts. In max (Luckhardt and Irani, 1986), the extension of minimax to multi-player games, pruning is not as successful. This paper focus on pruning techniques. There are many open questions in multi-player games, and we cannot cover them all here. For instance, it is unclear what the ‘best’ practical backup rule is. The techniques presented in this paper represent just one way we can evaluate the effectiveness of an algorithm. We first review the max algorithm and the conditions under which pruning can be applied to max. Based on this, we show that shallow pruning in max cannot occur in many multi-player games. We will examine another common pruning method, branch-and-bound pruning, showing that it faces the same limitations as alpha-beta pruning when applied to max trees. Finally, we present a hybrid algorithm, alphabeta branch-and-bound, which combines these two pruning techniques in multi-player games for more effective pruning. We will also analyze the reduction of a n-player game to a 2-player game. Examples: Hearts and Sergeant Major (8-5-3) To help make the concepts in this paper more clear, we chose two card games, Hearts and Sergeant Major, to highlight the successes and failures of the various algorithms presented. Note that while the game of bridge is played with 4 players, each player has the goal of maximizing the joint score they share with their partner, so bridge is really a two-team game, and standard minimax applies. Hearts and Sergeant Major, also known as 8-5-3, are both trick-based card games. That is, the first player plays (leads) a card face-up on the table, and the other players follow in order, playing the same suit if possible. When all players have played, the player who played the highest card in the suit that was led “wins” or “takes” the trick. He then places the played cards in his discard pile, and leads the next trick. This continues until all cards have been played. Cards are dealt out to each player before the game begins, and each game has special rules about passing cards between players before starting. Card passing has no bearing on the work presented here, so we ignore it. Hearts is usually played with four players, but there are variations for playing with three or more players. The goal of Hearts is to take as few points as possible. Each card in the suit of hearts is worth one point, and the queen of spades is worth 13. A player takes points when he takes a trick which contains point cards. At the end of the game, the sum of all scores is always 26, and each player can score between 0 and 26. If a player takes 26 points, or “shoots the moon,” the other players all get 26 points each. For now, we ignore this rule. Sergeant Major is a three-player game. Each player is dealt 16 cards, and the remainder of the deck is set aside. The ultimate goal for each player is to take as many tricks as possible. Similar to Hearts, the sum of scores is always 16, and each individual player can get any score from 0 to 16. More in-depth descriptions of these and other games mentioned here can be found in (Hoyle et al. 1991). On Pruning Techniques for Multi-Player Games Nathan R. Sturtevant and Richard E. Korf Computer Science Department University of California, Los Angeles Los Angeles, CA 90024 {nathanst, korf}@cs.ucla.edu Copyright © 2000, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. From: AAAI-00 Proceedings. Copyright © 2000, AAAI (www.aaai.org). All rights reserved.