Using Combinatorial Solutions and Atomic Weights to Play Competitive AI Clobber. Clobber is a twoplayer abstractstrategy game with no element of random chance. Introduced by game theorists Michael H. Albert, J. P. Grossman and Richard Nowakowski in 2001, it has subsequently been extensively studied for its combinatorial properties. Since 2005, it has been a major event in the Computer Olympiad.Although the concept of endgame tables has been introduced for Clobber and a use case suggested, usage of such has been scant in the Computer Olympiad or in the further development of other areas of Combinatorial Game Theory. We remedy this gap in this paper and demonstrate how to implement and use combinatorial solutions and atomic weights in an endgame table to perfect the play of Clobber by AI.We find that with the proper leveraging of endgame tables, it is possible to play perfectly on many Clobber positions of up to 70 pieces (on a 10x10 opening position) within time limits of most tournaments, and the solutions of Clobber on some significantly sized boards are likely reachable in the near future.

The subset sum game revisited We discuss a game theoretic variant of the subset sum problem, in which two players compete for a common resource represented by a knapsack. Each player owns a private set of items, players pack items alternately, and each player either wants to maximize the total weight of his own items packed into the knapsack or to minimize the total weight of the items of the other player. We show that finding the best packing strategy against a hostile or a selfish adversary is PSPACEcomplete, and that against these adversaries the optimal reachable item weight for a player cannot be approximated within any constant factor (unless PNP). The game becomes easier when the adversary is shortsighted and plays greedily: finding the best packing strategy against a greedy adversary is NPcomplete in the weak sense. This variant forms one of the rare examples of pseudopolynomially solvable problems that have a PTAS, but do not allow an FPTAS.