Maxn 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 maxn 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 maxn 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 CEparanoid 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 branch-and-bound. Using heuristic bounds in the 2-player reduction reduced node expansions another factor of 12.